Skew Continuous Morphisms of Ordered Lattice Ringoids

Skew continuous morphisms of ordered lattice semirings and ringoids are studied. Different associative semirings and non-associative ringoids are considered. Theorems about properties of skew morphisms are proved. Examples are given. One of the main similarities between them is related to cones in algebras of non locally compact groups.


Introduction
Semirings, ringoids, algebroids and non-associative algebras play important role in algebra and among them ordered semirings and lattices as well [1][2][3][4][5][6][7][8].This is also motivated by idempotent mathematical physics naturally appearing in quantum mechanics and quantum field theory (see, for example, [9] and references therein).They also arise from the consideration of algebroids and ringoids associated with non locally compact groups.Namely, this appears while the studies of representations of non locally compact groups, quasi-invariant measures on them and convolution algebras of functions and measures on them [10][11][12][13].The background for this is A. Weil's theorem (see [14]) asserting that if a topological group has a quasi-invariant σ-additive non trivial measure relative to the entire group, then it is locally compact.Therefore, it appears natural to study inverse mapping systems of non locally compact groups and their dense subgroups.Such spectra lead to structures of algebroids and ringoids.Investigations of such objects are also important for making advances in representation theory of non locally compact groups.
In this paper methods of categorial topology are used (see [15][16][17][18] and references therein).This article is devoted to ordered ringoids and semirings with an additional lattice structure.Their continuous morphisms are investigated in Section 3. Preliminaries are given in Section 2. Necessary definitions 2.1-2.4 are recalled.For a topological ringoid K and a completely regular topological space X new ringoids C(X, K) are studied, where C(X, K) consists of all continuous mappings f : X → K with point-wise algebraic operations.Their ideals, topological directed structures and idempotent operations are considered in Lemmas 2.6, 2.8, 2.9, 2.12 and Corollary 2.7.There are also given several examples 2.13-2.18 of objects.One of the main examples between them is related to cones in algebras of non locally compact groups.Another example is based on ordinals.Construction of ringoids with the help of inductive limits is also considered.
Structure and properties of these objects are described in Section 3. Definitions of morphisms of ordered semirings and some their preliminaries are described in Subsection 3.1.An existence of idempotent K-homogeneous morphisms under definite conditions is proved in Lemma 3.4.A relation between order preserving weakly additive morphisms and non-expanding morphisms is given in Lemma 3.7.An extension of an order preserving weakly additive morphism is considered in Lemma 3.9.
Then a weak* topology on a family O(X, K) of all order preserving weakly additive morphisms on a Hausdorff topological space X with values in K is taken.The weak* compactness of O(X, K) under definite conditions is proved in Theorem 3.10.Further in Proposition 3.11 there is proved that I(X, K) and I h (X, K) are closed in O(X, K), where I(X, K) denotes the set of all idempotent K-valued morphisms, also I h (X, K) denotes its subset of idempotent homogeneous morphisms.
Categories related to morphisms and ringoids are presented in Subsection 3.2.An existence of covariant functors, their ranges and continuity of morphisms are studied in Lemmas 3.14, 3.16, 3.21, 3.34 and Propositions 3.15, 3.22.In Propositions 3.24, 3.26 and 3.29 such properties of functors as being monomorphic and epimorphic are investigated.Supports of functors are studied in Proposition 3.31.Moreover, in Proposition 3.32 it is proved that definite functors preserve intersections of closed subsets.Then functors for inverse systems are described in Proposition 3.33.Bi-functors preserving pre-images are considered in Proposition 3.35.Monads in certain categories are investigated in Theorem 3.38.Exact sequences in categories are considered in Proposition 3.39.
Lattices associated with actions of groupoids on topological spaces are investigated in subsection 3.3.Supports of (T, G)-invariant semi-idempotent continuous morphisms are estimated in Proposition 3.42, where G is a topological groupoid and T is its representation described in Lemma 3.40.Structures of families of all semi-idempotent continuous morphisms associated with a groupoid G and a ringoid K are investigated in Proposition 3.43 and Theorems 3.44, 3.45.
The main results are Propositions 3.22, 3.24, 3.29, 3.32, 3.33, 3.35, 3.39, 3.43, Theorems 3.38, 3.44 and 3.45.All main results of this paper are obtained for the first time.The obtained results can be used for further studies of such objects, their classes and classification.They can be applied to investigations of non locally compact group algebras also.

Preliminaries
To avoid misunderstandings we first present our definitions.1. Definitions.Let K be a set and let two operations + : K 2 → K the addition and × : K 2 → K the multiplication be given so that K is a semigroup (with associative binary operations) or a quasigroup (with may be non-associative binary operations) relative to + and × with neutral elements e + =: 0 and e × =: 1 so that a × 0 = 0 × a = 0 for each a ∈ K and either the right distributivity a(b + c) = ab + ac for every a, b, c ∈ K or the left distributivity (b + c)a = ba + ca for every a, b, c ∈ K is accomplished, then K is called a semiring or a ringoid respectively with either right or left distributivity correspondingly.If it is simultaneously right and left distributive, then it is called simply a semiring or a ringoid respectively.
A semiring K (or a ringoid, or a ring, or a non-associative ring) having also a structure of a linear space over a field F and such that α(a + b) = αa + αb, 1a = a, α(ab) = (αa)b = a(αb) and (αβ)a = α(βa) for each α, β ∈ F and a, b ∈ K is called a semialgebra (or an algebroid, or an algebra or a non-associative algebra correspondingly).
A semiring K (or a semialgebra and so on) supplied with a topology on K (or on K and F correspondingly) relative to which algebraic operations are continuous is called a topological semiring (or a topological semialgebra and so forth correspondingly).
A set K with binary operations µ 1 , ..., µ n will also be called an algebraic object.An algebraic object is commutative relative to an operation µ p if µ p (a, b) = µ p (b, a) for each a, b ∈ K.
An algebraic object K with binary operations µ 1 , ..., µ n is called either directed or linearly ordered or well-ordered if it is such as a set correspondingly and its binary operations preserve an ordering: µ p (a, b) ≤ µ p (c, d) for each p = 1, ..., n and for every a, b, c, d ∈ K so that a ≤ c and b ≤ d when a, b, c, d belong to the same linearly ordered set Z in K.
Henceforward, we suppose that the minimal element in an ordered K is zero.Henceforth, for semialgebras, non-associative algebras or algebroids A speaking about ordering on them we mean that only their non-negative cones K = {y : y ∈ A, 0 ≤ y} are considered.For non-negative cones K in semialgebras, non-associative algebras or algebroids only the case over the real field will be considered.
2. Definition.A (non-associative) topological algebra or a topological ringoid, etc., we call topologically simple if it does not contain closed ideals different from {0} and K, where K = {0}.
3. Definition.We consider a directed set K which satisfies the condition: (DW) for each linearly ordered subset A in K there exists a well-ordered subset B in K such that A ⊂ B.

Definitions.
Let K be a well-ordered (or directed satisfying condition 3(DW)) either semiring or ringoid (or a non-negative cone in a algebroid over the real field R) such that (1) sup E ∈ K for each E ∈ T, where T is a family of subsets of K.
If K is a directed topological either semiring or ringoid, we shall suppose that it is supplied with a topology (2) τ = τ K so that every set That is, if a set Z is linearly ordered in K this topology τ K provides the hereditary topology on Z which is not weaker than the interval topology on Z generated by the base For a completely regular topological space X and a topological semiring (or ringoid) K let C(X, K) denote a semiring (or a ringoid respectively) of all continuous mappings f : X → K with the element-wise addition ( f + g)(x) = f (x) + g(x) and the element-wise multiplication ( f g)(x) = f (x)g(x) operations for every f , g ∈ C(X, K) and x ∈ X.
If K is a directed semiring (or a directed ringoid) and X is a linearly ordered set, C + (X, K) (or C − (X, K)) will denote the set of all monotone non-decreasing (or non-increasing correspondingly) maps f ∈ C(X, K).
For the space C(X, K) (or For each non-void set A in On there exists sup A ∈ On (see [20]).
If K is a linearly ordered non-commutative relative to the addition semiring (or a ringoid), then the new operation (a, b) → max(a, b) =: a + 2 b defines the commutative addition.Then c(a + 2 b) = max(ca, cb) = ca + 2 cb and (a + 2 b)c = max(ac, bc) = ac + 2 bc for every a, b, c ∈ K, that is (T, + 2 , ×) is left and right distributive.
As an example of a semiring (or a ringoid) K in Definitions 4 one can take K = On or where a < b ∈ On.Evidently, K = On satisfies Condition 4(1), since sup E exists for each set E in On (see [20]).
Particularly, if a topological space X is compact and C(X, K) is a semiring (or a ringoid) of all continuous mappings f : X → K, then a family T contains the family of compact subsets { f (X) : f ∈ C(X, K)}, since a continuous image of a compact space is compact (see Theorem 3.1.10 [21]).
It is possible to modify Definition 4 in the following manner.For a well-ordered K without Condition 4(1) one can take the family of all continuous bounded functions f : X → K and denote this family of functions by C(X, K) for the uniformity of the notation.
For a directed K satisfying Condition 3(DW) without Condition 4(1) it is possible to take the family of all monotone non-decreasing (or non-increasing) bounded functions f : X → K for a linearly ordered set X and denote this family by C + (X, K) ( C − (X, K) correspondingly) also.
Naturally, C(X, K) has also the structure of the left and right module over the semiring (or the ringoid correspondingly) K, i.e., a f and f a belong to C(X, K) for each a ∈ K and f ∈ C(X, K).To any element a ∈ K the constant mapping g a ∈ C(X, K) corresponds such that g a (x) = a for each x ∈ X.
The semiring (or the ringoid) C(X, K) will be considered directed: (1) f ≤ g if and only if f (x) ≤ g(x) for each x ∈ X. (1).Then there exists c ∈ K so that a ≤ c and b ≤ c, consequently, f ≤ g c and h ≤ g c .Thus for each f , h ∈ C(X, K) there exists q ∈ C(X, K) so that f ≤ q and h ≤ q.From a + b ≤ c + d and ab ≤ cd for each a ≤ c and b ≤ d in K it follows that f + q ≤ g + h and f q ≤ gh for each f ≤ g and q ≤ h in C(X, K).
Henceforth, we consider cases, when (3) a topology on X is sufficiently fine so that functions separate points in X, i.e., for each x = z in X there exists f in C(X, K)(or C − (X, K) or C + (X, K) correspondingly) such that f (x) = f (z).
The latter is always accomplished in the purely algebraic discrete case.
If E is a closed subspace in a topological space X, then C(X, K|E) Proof.For a clopen topological subspace E in X one gets C(E, K) isomorphic with C(X, K|E), since each f ∈ C(E, K) has the zero extension on X \ E.
8. Lemma.For a linearly ordered set X and a directed semiring (ringoid) K there are directed semirings (or ringoids correspondingly) C + (X, K) and C − (X, K).
Proof.The sets C + (X, K) and C − (X, K) are directed according to Condition 5(1) with a partial ordering inherited from C(X, K).Since a + b ≤ c + d and ab ≤ cd for each a ≤ c and b ≤ d in K, then f + q ≤ g + h and f q ≤ gh for each f ≤ g and q ≤ h all either in C + (X, K) or C − (X, K).On the other hand, for each f , h ∈ C(X, K) there exists g c ∈ C(X, K) so that f ≤ g c and h ≤ g c (see § 5).If f (x) ≤ f (y) and h(x) ≤ h(y) for f , h ∈ C + (X, K) and each x ≤ y in K, then f (x) + h(x) ≤ f (y) + h(y) and f (x)h(x) ≤ f (y)h(y), consequently, f + h and f h are in C + (X, K).Analogously, if f , h ∈ C − (X, K), then f + h and f h are in C − (X, K).But a constant mapping g c belongs to C + (X, K) and C − (X, K).Thus C + (X, K) and C − (X, K) are directed semirings (or ringoids correspondingly).9. Lemma.If H = H X is a covering of X and τ K is a topology on K satisfying Conditions 3(DW) and 4(1 − 3), then a semiring (or ringoid or a non-negative cone in a algebroid over R) C(X, K) can be supplied with a topology relative to which it is a topological directed (TD) semiring (or a TD ringoid or a TD algebroid respectively).
Proof.Take a topology τ C on C(X, K) with the base β C formed by the following sets and their finite intersections: where g ∈ C(X, K), A ∈ H, V ∈ τ K .Evidently, the addition + = µ 1 and the multiplication × = µ 2 are continuous relative to this topology, since each U ∈ τ C is the union of base sets P ∈ β C .In view of Section 5 C(X, K) is directed: µ p ( f , h) ≤ µ p (g, u) for each p = 1, 2 and for every f , g, h, u ∈ C(X, K) so that f ≤ g and h ≤ u when f , g, h, u belong to the same linearly ordered set in C(X, K), since element-wise these inequalities are satisfied in K, i.e. for f (x), g(x), h(x), u(x) with x ∈ X (see §1).
10. Note.Henceforward, it will be supposed that C(X, K) is supplied with the topology τ C of Lemma 9, while C + (X, K) and C − (X, K) are considered relative to the topology inherited from C(X, K).Particularly, if X ∈ H X , then it provides the topology of the uniform convergence on C(X, K).
11. Corollary.If the conditions of Lemma 9 are satisfied and H = 2 X is the family of all subsets in X and a topology τ K on K is discrete, then τ C is the discrete topology on C(X, K).
12. Lemma.Suppose that the conditions of Lemma 9 are satisfied.Then the functions (1) f ∨ g(x) := max( f (x), g(x)) and (2) f ∧ g(x) := min( f (x), g(x)) are in C(X, K) (or in C + (X, K) or in C − (X, K)) for every pair of functions f , g ∈ C(X, K) (or in C + (X, K) or in C − (X, K) correspondingly) satisfying the condition: (3) for each x ∈ X either f (x) < g(x) or g(x) < f (x) or f (x) = g(x).
Proof.Let f , g ∈ C(X, K) satisfy Condition (3).Then the sets {x : x ∈ X, f (x) ≤ g(x)} and {x : x ∈ X, f (x) ≤ g(x)} are closed in X, since f and g are continuous functions on X and the topology τ K on K satisfies Condition 4(2).For each closed set E in K the sets Relative to the topology of §9 on C(X, K) operations ∨ and ∧ are continuous on C(X, K), C + (X, K) and C − (X, K).

Example. Ringoids and ordinals.
The class On of all ordinals has the addition µ 1 = + o and the multiplication µ 2 = × o operations which are generally non-commutative, associative, with unit elements 0 and 1 respectively, on On the right distributivity is satisfied (see  and Examples 1-3 in [19,22]).Relative to the interval topology generated by the base {(a, b) : a < b ∈ On} the class On is the topological well-ordered semiring, where (a, b) = {c : c ∈ On, a < c < b}.For each non-void set A in On there exists sup A ∈ On (see [20]).
14. Example.Construction of ringoids with the help of inductive limits.Let J be a directed set of the cardinality card(J) ≥ ℵ 0 such that for each l, k ∈ J there exists j ∈ J with l ≤ j and k ≤ j (see also §I.3 [21]), and let φ : J → J be a monotone decreasing map, G j ⊆ [0, ∞), let also p k j : G k → G j be an embedding for each k ≤ j ∈ J.There is considered G j as a ringoid with the addition, the multiplication, with neutral elements 0 j = 0 by addition and 1 j = 1 by multiplication and the linear ordering x j < y j inherited from [0, ∞) = {t : t ∈ R, 0 ≤ t < ∞} for each j ∈ J. Put G 0 = lim{G j , p k j , J} to be the inductive limit of the direct mapping system so that G is the quotient ( j G j )/Ξ of the direct sum j G j by the equivalence relation Ξ caused by mappings p k j .Then consider G := {x : x ∈ G 0 , sup j∈J x j < ∞}, where x j = p j (x), p j : G → G j notates the projection.
Then we define g + 1 h := {v j : v j = g j + h j ∀j ∈ J} and g × 1 h := {w j : w j = g j p k j (h k )∀j ∈ J with k = φ(j)} for all g, h ∈ G, where g j = p j (g) for each j ∈ J. Let also x < 1 y in G if and only if x j < y j for each j ∈ J. Certainly for each x, y ∈ G there exists z ∈ G so that x ≤ 1 z and y ≤ 1 z, for example, z j = max(x j , y j ) for each j ∈ J. Therefore we get that if x < 1 y and u < 1 z in G, then x + 1 u < 1 y + 1 z and x × 1 u < 1 y × 1 z.We supply G with a topology τ b inherited from the inductive limit topology on G 0 , where [0, ∞) is supplied with the standard metric of R and G j has the topology inherited from [0, ∞).Then we deduce that U(x j , b, j) + U(z j , b, j) ⊂ U(x j + z j , 2b, j) and U(x j , b, j)U(z k , b, k) ⊂ U(x j p k j (z k ), b(1 + x j + z k ), j) for every x, z ∈ G and b > 0 and j ∈ J with k = φ(j), where U(x j , b, j) := {y j : y j ∈ G j , x j − b < y j < x j + b}.Since sup j∈J x j < ∞ for each x ∈ G, then the addition and the multiplication in G are continuous.Thus (G, + 1 , × 1 , < 1 , τ b ) is the topological directed ringoid with the left and the right distributivity in which the multiplication × 1 is non-associative, since φ(j) < j for each j ∈ J.It is worth to note that each set of the form S(x) := {y : y ∈ G, either x < y or x is incomparable with y} is open in (G, τ b ), where x ∈ G.
15. Example.The case of G j ⊆ [0, ∞) ω for each j ∈ J, where ω is a directed set, can be considered analogously to Example 14, taking the lexicographic ordering on the Cartesian product M := ω × J and considering M instead of J.
16. Example.On G from Example 14 one can take also x + 2 y := {v j : v j = max(x j , y j )∀j ∈ J} and x × 2 y := {w j : w j = min(x j , y k )∀j ∈ J with k = φ(j)}.Then (G, + 2 , × 2 , < 1 , τ b ) is a topological non-associative ringoid with the left and right distributivity.17.Example.Ringoids associated with families of measures.Let G j be a Boolean algebra on a set H j and let p k j : G k → G j be an embedding for each j where J and φ are as in subsection 14.Suppose that on each Boolean algebra G j there is a probability (finitely additive) measure m j : Consider on G the inductive limit topology τ b , where G j is supplied with the metric d j for each j ∈ J. Naturally it is possible to put A ≤ B in G if and only if A j ⊆ B j for each j ∈ J. Then the inequalities m j (( is the topological ringoid with the left and right distributivity and the non-associative multiplication.Instead of measures it is possible more generally to consider submeasures m j , that is possessing the subadditivity property: 18. Example.Ringoids induced by spectra of non locally compact groups.Let {G j , p k j , J} be a family of topological non locally compact groups G j , where J is a directed set, p k j : G k → G j is a continuous injective homomorphism for each j < k in J. Let also φ : J → J be an increasing map and let m j : B j → [0, 1] be a Radon probability σ-additive measure on the Borel σ-algebra B j of G j such that m j is left quasi-invariant relative to p k j (G k ) for each j ∈ J with k = φ(j).That is there exists the Radon-Nikodym derivative (i.e., the left quasi-invariance factor) . Suppose also that there exists an open base of neighborhoods of e k ∈ G k such that their closures in G j are compact.
It is known that such systems exist for loop groups and groups of diffeomorphisms and Banach-Lie groups. Then There exists the non-associative normed algebra [11][12][13]23]).Now we take the positive cone If f , h, q, u ∈ F and f ≤ q, h ≤ u, then f j (g j ) + h j (g j ) ≤ q j (g j ) + u j (g j ) and for m j -almost all g j ∈ G j and hence f + h ≤ q + u and f × h ≤ q × u.Then we infer that For each f , h ∈ F there exists an element u ∈ F so that f ≤ u and h ≤ u, for example, either u = f + h or u given by the formula u j (g j ) = max( f j (g j ), h j (g j )) for each j ∈ J and m j -almost all g j ∈ G j .
Take on F the topology τ n inherited from the norm topology on E .This implies that (F, +, ×, < , τ n ) is the directed topological non-associative ringoid with the left and right distributivity.
There is the decomposition If f and h in F are incomparable, there exist j, l ∈ J (may be either j = l or j = l) such that m j (A + j ) > 0 and m l (A − l ) > 0, where for each q ∈ E and j ∈ J, while m j is the probability measure for each j ∈ J. On the other hand, if v < u in F, there exists l ∈ J so that m l (A − l (v, u)) > 0 and v j (g j ) ≤ u j (g j ) for m j -almost all g j ∈ G j for each j ∈ J. Therefore, for 0 < b prescribed by the inequality given above and each q ∈ B(F, h, b) the inequality q ≤ f is impossible, consequently, either q is incomparable with f or f < q.Thus each set of the form S( f 19. Note.Certainly relative to the discrete topology the aforementioned ringoids are also topological ringoids.Other examples can be constructed from these using the theorems and the propositions presented above.

Morphisms and Their Properties
2. Definition.We call a mapping ν on C(X, K) (or A mapping (morphism) ν on C(X, K) (or C + (X, K) or C − (X, K)) with values in K we call order preserving (non-decreasing), if (6) A morphism ν will be called K-homogeneous on C(X, K) (or 3. Remark.If a morphism satisfies Condition 2(4), then it is order preserving.The evaluation at a point morphism δ x defined by the formula: (1) If morphisms ν 1 , ..., ν n are idempotent and the multiplication in K is distributive, then for each constants (2) are idempotent.Moreover, if the multiplication in K is commutative, associative and distributive and constants satisfy Conditions (2, 3) and morphisms ν 1 , ..., ν n are K-homogeneous, then morphisms of the form (4, 5) are also K-homogeneous.
The considered here theory is different from the usual real field R, since R has neither an infimum nor a supremum, i.e. it is not well-ordered and satisfy neither 2.3(DW) nor 2.4(1).

Lemma. Suppose that either
(1) K is well-ordered and satisfies Conditions 2.4(1 − 3) or (2) X is linearly ordered and K is directed and satisfies Conditions 2.3(DW) and 2.4(1 − 3).Then there exists an idempotent K-homogeneous morphism ν on C(X, K) in case (1), on C + (X, K) and C − (X, K) in case (2).Moreover, if K ⊂ On and K is infinite, X is not a singleton, ℵ 0 ≤ |K|, |X| > 1, then ν has not the form either 3(4) or 3 (5) with the evaluation at a point morphisms ν 1 , ..., ν n relative to the standard addition in On.
Proof.Suppose that ν is an order preserving morphism on C(X, K) (or (3), then in accordance with Lemma 2.12 there exists f ∨ g and f ∧ g in the corresponding C(X, K) (or Let also E be a subset in X, we put . This morphism exists due Conditions 2.4(1, 3), since in both cases (1) and (2) of this lemma, the image f (E) is linearly ordered and is contained in K.
From the fact that the addition preserves ordering on K (see §2.1) it follows that Properties (1 − 3, 7, 8) are satisfied for the morphism ν given by Formula (3).If f ≤ g on X, then for each a ∈ f (E) there exists b ∈ g(E) so that a ≤ b, consequently, ν( f ) ≤ ν(g), i.e., 2(6) is fulfilled.
We consider any pair of functions f , g in C(X, K) (or C + (X, K) or C − (X, K)) satisfying Condition 8(3).In case (2) a topological space X is linearly ordered, in case (1) K is well-ordered, hence f (X), g(X), f (E) and g(E) are linearly ordered in K. Then for each a ∈ Thus Properties 2(4, 5) are satisfied as well.
If E is chosen such that there exists U ∈ H X with E ⊂ U, then this morphism ν is continuous on C(X, K), C + (X, K) and C − (X, K) (see § §2.3, 2.4, 2.9 and 2.10 also).
If a set X is not a singleton, |X| > 1, and K ⊂ On is infinite, ℵ 0 ≤ |K|, then taking a set E in X different from a singleton, |E| > 1, we get that the morphism given by Formula (3) can not be presented with the help of evaluation at a point morphisms ν 1 = δ x 1 , ..., ν n = δ x n by Formula either 3(4) or 3 (5) relative to the standard addition in On, since functions f in C(X, K) (or C + (X, K) or C − (X, K)) separate points in X (see Remark 2.5(3)).
5. Remark.Relative to the idempotent addition x ∨ y = max(x, y) the morphism ν E given by 4(3) has the form ν Let I(X, K) denote the set of all idempotent K-valued morphisms, while I h (X, K) denotes its subset of idempotent homogeneous morphisms.
A set F of all continuous K-valued morphisms on C(X, K) is supplied with the weak* topology having the base consisting of the sets (1) where ν may be non-linear or discontinuous as well.
The family of all order preserving weakly additive morphisms on a Hausdorff topological space X with values in K will be denoted by O(X, K).
If E ⊂ C(X, K) satisfies the conditions: g 0 ∈ E, g + b and b + g ∈ E for each g ∈ E and b ∈ K, then E is called an A-subset.
7. Lemma.If ν : C(X, K) → K is an order preserving weakly additive morphism, then it is non-expanding.
Proof.Suppose that f , h ∈ C(X, K) and b ∈ K are such that f (x) Thus the morphism ν is non-expanding.
8. Corollary.Suppose that a topological ringoid K is well-ordered, satisfies 1(1) and with the interval topology, X ∈ H, C(X, K) is supplied with the topology of §2.9.Then any order preserving weakly additive morphism ν : C(X, K) → K is continuous.
Proof.This follows from Lemma 7 and § §2.3, 2.4, since each subset { f : f ≤ g} and { f : g ≤ f } is closed in C(X, K) in the topology of §2.9, where g ∈ C(X, K). 9. Lemma.Suppose that A is an A-subset (a left or right submodule over K) in C(X, K) and ν : A → K is an order preserving weakly additive morphism (left or right K-homogeneous with left or right distributive ringoid K correspondingly).Then there exists an order preserving weakly additive morphism µ : C(X, K) → K such that its restriction on A coincides with ν.
Proof.One can consider the set F of all pairs (B, µ) so that B is an A-subset (a left or right submodule over K respectively), A ⊆ B ⊆ C(X, K), µ is an order preserving weakly additive morphism on B the restriction of which on A coincides with ν.The set F is partially ordered: and µ 2 is an extension of µ 1 .In accordance with Zorn's lemma a maximal element (E, µ) in F exists.
due to Conditions 2.3(DW) and 2.4(1) imposed on K. Then we put F = E ∪ {g + g c , g c + g : c ∈ K} ( F is a minimal left or right module over K containing E and g correspondingly).Then one can put µ(g + g c ) = b + c and µ(g c + g) = c + b.Moreover, one gets µ(d(g + g c )) = dµ(g) + dc or µ((g + g c )d) = µ(g)d + cd for each d ∈ K correspondingly for each c ∈ K. Then µ is an order preserving weakly additive morphism (left or right homogeneous correspondingly) on F. This contradicts the maximality of A.
10. Theorem.If a ringoid K is well-ordered and satisfies 1(1), with the interval topology and K is locally compact, X ∈ H X .Then O(X, K) is compact relative to the weak* topology.
Proof.In view of Lemma 8 each ν ∈ O(X, K) is continuous.The set O(X, K) is supplied with the weak* topology (see §5).
For each ν ∈ O(X, K) one has ν(g c ) = ν(g c + g 0 ) = c, since g c + g 0 = g c and ν(g c + g 0 ) = c + 0. On the other hand, for each g ∈ C(X, K) due to Condition 2.4(1) a supremum exists, g := sup x∈X g(x) ∈ K.Each segment [a, b] in K is closed, bounded and hence compact relative to the interval topology.Therefore, O(X, K) is contained in the Tychonoff product S = ∏{[0, g ] : g ∈ C(X, K)}, since g ≤ h and hence ν(g) ≤ ν(h) when h(x) = g for each x ∈ X.This product is compact as the Tychonoff product of compact topological spaces by Theorem 3.2.13[21].It remains to prove, that O(X, K) is closed in S, since a closed subspace of a compact topological space is compact (see Theorem 3.1.2[21]).
Each compact Hausdorff space has a uniformity compatible with its topology (see Theorems 3.19 and 8.1.20[21]).To each element y ∈ S a morphism y : (2,3,6) for each ν n imply Properties 2(2, 3, 6) for q, since each segment [a, b] in K is compact and hence complete as the uniform space due to Theorem 8.3.15[21], where a < b ∈ K. Therefore, lim n = q ∈ O(X, K) according to Lemma 7 and Corollary 8. Thus O(X, K) is complete as the uniform space by Theorem 8.3.20 [21] and hence closed in S in accordance with Theorem 8.3.6 [21].
11. Proposition.In the topological space O(X, K) the subsets I(X, K) and I h (X, K) are closed.
12. Corollary.If the conditions of Theorem 10 are satisfied, then the topological spaces I(X, K) and I h (X, K) are compact.

Categories of Semirings, Ringoids and Morphisms
By I( f ) will be denoted the restriction of O( f ) onto I(X, K).
A T 1 topological space will be called K-completely regular (or K Tychonoff space), if for each closed subset F in X and each point x ∈ X \ F a continuous function h : X → K exists such that h(x) = 0 and h(F) = {c}, i.e. h is constant on F, where c = 0.
Let RK denote a category such that a family Ob(RK) of its objects consists of all K-regular topological spaces, a set of morphisms Mor(X, Y) consists of all continuous mappings f : X → Y for every X, Y ∈ Ob(RK), i.e.RK is a subcategory in the category of topological spaces.We denote by OK a category with objects Ob(OK) = {O(X, K) : X ∈ Ob(RK)} and families of morphisms Mor(O(X, K), O(Y, K)).

Lemma. (1).
There exists a covariant functor O in the category RK. (2).Moreover, if a topological ringoid K is well-ordered, satisfies 2.4(1) and with the interval topology, when (2).If ν j is a net converging to ν in O(X, K) relative to the weak* topology, then
If for each f ∈ Mor(X, Y) and each closed subset A in Y, the equality (F( f ) −1 )(F(A)) = F( f −1 (A)) is satisfied, then a covariant functor F is called preimage-preserving.When F( j∈J X j ) = j∈J F(X j ) for each family {X j : j ∈ J} of closed subsets in X ∈ Ob(RK) the monomorphic functor F is called intersection-preserving.
If a functor F preserves inverse mapping system limits, it is called continuous.A functor F is said to be weight-preserving when w(X) = w(F(X)) for each X ∈ Ob(RK), where w(X) denotes the topological weight of X ∈ Ob(RK).
A functor is said to be semi-normal when it is continuous, monomorphic, epimorphic, preserves weights, intersections, preimages and the empty space.
If a functor is continuous, monomorphic, epimorphic, preserves weights, intersections and the empty space, then it is called weakly semi-normal.17.Lemma.Let Y be a normal topological space, let also A and B be nonintersecting closed subsets in Y, where T is a well-ordered set supplied with the interval topology.Suppose also that c where d(X) denotes the density of a topological space X, |E| denotes the cardinality of E. There exist open subsets U and V in X such that (2 where cl X G denotes the closure of a set G in X. Sets V t will be defined by the transfinite induction.For this one can put In view of the Zermelo theorem there exists an ordinal P such that |P| = |E|, a bijective surjective mapping θ : P → E exists such that inf P = 0, 1 ∈ P, θ(0) = c 1 and θ(1) = c 2 .Suppose that V t j satisfying Condition (3) are constructed for j = 1, ..., n, j ∈ P.There exist elements a n = inf{t j : j ≤ n, t j < t n+1 } and b n = sup{t j : j ≤ n, t j < t n+1 }.Therefore, cl X V a n ⊂ V b n .From the normality of X it follows that open sets U and V exist such that cl Then one puts V t n+1 = U.This means that there exists a countable infinite sequence V t j for j ∈ ω 0 ⊆ P satisfying Conditions (3,4).If {t j : j ∈ ω 0 } is not dense in [c 1 , c 2 ] the process continues.Suppose that α is an ordinal such that ω 0 ⊆ α ⊂ P, V t j is defined for each j ∈ α.If the set {t j : j ∈ α} is not dense in [c 1 , c 2 ], there exists a segment (5 We put L = t j <a;j∈α V t j and M = b<t j ;j∈α V t j .From (3,4) it follows that the set L is open in X and L ⊂ M. On the other hand, ( The family F = {(V j : j ∈ α) : α ⊂ P} is ordered by inclusion: (V j : j ∈ α) ≤ (W k : k ∈ β) if and only if a bijective monotonously increasing mapping θ : α → β exists such that V j = W θ(j) for each j ∈ α.If a subfamily {(V j : j ∈ α) : α k ⊂ P, k ∈ Λ} is linearly ordered, then its union is in F .In view of the Kuratowski-Zorn lemma there exists a maximal element (V j : j ∈ α 1 ) in F for some ordinal α 1 ⊂ P such that conditions (3,4) (3,4), also from (6, 7) when (5) is fulfilled, and the definition of f it follows that 18. Lemma.If X is well-ordered and E is a segment [a, b] in X, while K satisfies Condition 2.3(DW), then each f ∈ C + (E, K) has a continuous extension g ∈ C + (X, K).
Proof.Since f (E) =: A is linearly ordered in K, then by 2.3(DW) there exists a well ordered subset B in K such that A ⊂ B. So putting g(x) = inf A for each x < a in X, whilst g(x) = sup A for each b < x in X one gets the continuous extension g ∈ C + (X, K) of f , that is g| E (y) = f (y) for each y ∈ E, since inf A and sup A exist in K due to 2.3(DW) and 2.4(1).
19. Definition.It will be said that a pair (X, K) of a topological space X and a ringoid K has property (CE) if for each closed subset E in X and each continuous function Henceforward, it will be supposed that a pair (X, K) has property (CE).20.Definitions.If Hausdorff topological spaces X and Y are given and f : X → Y is a continuous mapping, K 1 , K 2 are ordered topological ringoids (or may be particularly semirings) with an order-preserving continuous algebraic homomorphism u : By I( f , u) will be denoted the restriction of O( f , u) onto I(X, K).The shorter notations O( f ) and I( f ) are used when K is fixed, i.e. u = id.When X = Y and f = id we write simply O 2 (u) and I 2 (u) respectively omitting f = id.
Let S denote a category such that a family Ob(S ) of its objects consists of all topological spaces, a family of morphisms Mor(X, Y) consists of all continuous mappings f : X → Y for every X, Y ∈ Ob(S ).
Let K be the category objects of which Ob(K) are all ordered topological ringoids satisfying Conditions 2.3 and 2.4, Mor(A, B) consists of all order-preserving continuous algebraic homomorphisms for each A, B ∈ K. Then by K w we denote its subcategory of well-ordered ringoids and their order-preserving algebraic continuous homomorphisms.
We denote by OK a category with the families of objects Ob(OK) = {O(X, K) : X ∈ Ob(S ), K ∈ Ob(K w )} and morphisms Mor(O(X, K 1 ), O(Y, K 2 )) for every X, Y ∈ Ob(S ) and K 1 , K 2 ∈ Ob(K w ).Furthermore, IK stands for a category with families of objects Ob(I K) = {I(X, K) : X ∈ Ob(S ), K ∈ Ob(K w )} and morphisms Mor(I(X, K 1 ), I(Y, K 2 )) for every X, Y ∈ Ob(S ) and By S l will be denoted a category objects of which are linearly ordered topological spaces, while Mor(X, Y) consists of all monotone nondecreasing continuous mappings f : Subcategories of left homogeneous continuous morphisms we denote by O h K, O l,h K, I h K, I l,h K correspondingly.These morphisms are taken on subcategories K w,l in K or K l in K of left distributive topological ringoids.
21. Lemma.There exist covariant functors O, O h and O l , O l,h in the categories S and S l respectively.Proof.Suppose that X, Y ∈ Ob(S ) and f ∈ Mor(X, Y), while g ≤ h in C(Y, K), where Then for 1 X ∈ Mor(X, X), that is 1 X (x) = x for each x ∈ X, one deduces 1 X • q = q for each q ∈ Mor(Y, X) and t • 1 X = t for each t ∈ Mor(X, Y).On the other hand, 23. Definitions.A covariant functor F : S → S will be called epimorphic (monomorphic) if it preserves continuous epimorphisms (monomorphisms).If φ : A → X is a continuous embedding, then F(A) will be identified with F(φ)(F(A)).
If for each f ∈ Mor(X, Y) and each closed subset A in Y, the equality (F( f ) −1 )(F(A)) = F( f −1 (A)) is satisfied, then a covariant functor F is called preimage-preserving.In the case F( j∈J X j ) = j∈J F(X j ) for each family {X j : j ∈ J} of closed subsets in X ∈ Ob(S ) (or in Ob(S l )), the monomorphic functor F is called intersection-preserving.
If a functor F preserves inverse mapping system limits, it is called continuous.
A functor is said to be semi-normal when it is monomorphic, epimorphic, also preserves intersections, preimages and the empty space.
If a functor is monomorphic, epimorphic, also preserves intersections and the empty space, then it is called weakly semi-normal.
24. Proposition.The functor O (or O h , O l , O l,h ) is monomorphic.
Proof.Let X, Y ∈ Ob(S ) (or in Ob(S l ) respectively) with a continuous embedding s : X → Y (order-preserving respectively).Then we suppose that 25. Corollary.The functors I, I h , I l and I l,h are monomorphic.
Proof.This follows from Proposition 24 and Definitions 20.

Proposition. The functors O, O
The set L of all continuous mappings g • f : X → K with g ∈ C(Y, K) (or in C + (Y, K) correspondingly) is the A-subset according to Definitions 6 or the left module over K in C(X, K) (or in C + (X, K)).Then we put µ(g • f ) = ν(g).This continuous morphism has an extension from L to a continuous morphism µ ∈ O(X, K) (or in O h (X, K), O l (X, K), O l,h (X, K) correspondingly) due to Lemmas 9, 14 and Corollary 8.
27. Lemma.Let L be a submodule over K of C(X, K) or C + (X, K) relative to the operations ∨, ∧, × 2 and containing all constant mappings g c : X → K, where c ∈ K. Let also ν : L → K be an idempotent (left homogeneous) continuous morphism.For each f ∈ C(X, K) \ L or C + (X, K) \ L there exists an idempotent (left homogeneous) continuous extension µ M of ν on a minimal closed submodule M containing L and f .
Proof.For each g ∈ M we put (1) On the other hand for each g 1 , From the inequalities 28. Lemma.If suppositions of Lemma 27 are satisfied, then there exists an idempotent (left homogeneous) continuous morphism λ on C(X, K) or C + (X, K) respectively such that λ| L = ν.
Proof.The family of all extensions (M, µ M ) of ν on closed submodules M of C(X, K) or C + (X, K) respectively is partially ordered by inclusion: (M, µ M ) ≤ (N, µ N ) if and only if M ⊂ N and ν N | M = ν M .In view of the Kuratowski-Zorn lemma [20] there exists the maximal closed submodule P in C(X, K) or C + (X, K) correspondingly and an idempotent extension ν P of ν on P. If P = C(X, K) or C + (X, K) correspondingly by Lemma 27 this morphism ν P could be extended on a module L containing P and some g ∈ C(X, K) \ P or in C(X, K) + \ P respectively.This contradicts the maximality of (P, ν P ).Thus P = C(X, K) or C + (X, K) correspondingly.
29. Proposition.The functors I, I l and I h , I l,h are epimorphic.
Proof.Let a continuous mapping f : X → Y be epimorphic.We consider the set L of all continuous mappings g • f : Then L is a submodule of C(X, K) or C + (X, K) relative to the operations ∨, ∧, × 2 and L contains all constant mappings g c : X → K, where c ∈ K. Then we put µ(g Moreover, E is a support of ν if and only if ν is supported on E and for each proper closed subset F in E, i.e.F ⊂ E with F = E, there are f , h ∈ C(X, K) or in C + (X, K) respectively with f | F ≡ h| F such that ν( f ) = ν(h).
Proof.Consider ν ∈ O(X, K) such that ν( f ) = ν(g) for each functions f , g : X → K with f | E = g| E .A continuous morphism ν induces a continuous morphism λ ∈ O(E, K) such that λ(h) = ν(h) for each h ∈ C(X, K) with h| X\E = 0. Denote by id the identity embedding of a closed subset E into X.Each function t : E → K has an extension on X with values in K by Condition 19(CE).Then O(id)(λ) = ν, since ν(g 0 ) = 0 and hence ν(s) = 0 for each s ∈ C(X, K) such that s| E ≡ 0.
If ν ∈ O(X, K) and ν is supported on E, then by Definition 30 there exists a morphism λ ∈ O(E, K) such that O(id)(λ) = ν.Therefore the equalities are valid: If E is a support of ν, then by the definition this implies that ν is supported on E. Suppose that If ν is supported on E and for each proper closed subset 32. Proposition.The functors O, I, O l , I l , O l,h , I l,h preserve intersections of closed subsets.
Proof.If E is a closed subset in X, then there is the natural embedding In view of Proposition 31 the functors O and O l preserve intersections of closed subsets.This implies that the functors I, I l , O l,h and I l,h also have this property.
33. Proposition.Let {X b ; p b a ; V} =: P be an inverse system of topological spaces X b , where V is a directed set, p b a : X b → X a is a continuous mapping for each a ≤ b ∈ V, p b : X = lim P → X b is a continuous projection.Then the mappings are bijective and surjective continuous algebraic homomorphisms.Moreover, if X b ∈ Ob(S l ) and p b a is order-preserving for each a < b ∈ V, then the mappings (3)  • p b = p a .Let q : O(X, K) → Y denote the limit map of the inverse mapping system q = lim{O(p a ); O(p a b ); V} (see also §2.5 [21]). A If ν, λ ∈ O(X, K) are two different continuous morphisms, then this means that a continuous function f ∈ C(X, K) exists such that ν 1 ( f ) = ν 2 ( f ).This is equivalent to the following: there exists a ∈ V such that (O(p a )(ν))( f ) = (O(p a )(λ))( f ).Thus the mappings s and analogously t are surjective and bijective.
On the other hand, (7) for each c ∈ K and f b ∈ C(X b , K). Taking the inverse limit in Equalities (5 − 10) gives the corresponding equalities for ν ∈ I(X, K), where ν = lim{ν a ; I(p b a ); V}, hence t is the continuous algebraic homomorphism due to Theorem 2.5.8 [21].

Lemma.
There exist covariant functors O 2 , I 2 , and O l,2 , I l,2 and O h,2 , I h,2 and O l,h,2 , I l,h,2 in the categories K w and K and K w,l and K l respectively. Proof.
If f (x) ≤ g(x), then u( f (x)) ≤ u(g(x)), where x ∈ X, f , g ∈ C(X, K 1 ).Therefore, if a mapping either f ∨ g or f ∧ g exists in C(X, K 1 ), then u( f This and the definitions above imply that O 2 (u) : O(X, K 1 ) → O(X, K 2 ), I 2 (u) : I(X, K 1 ) → I(X, K 2 ) and O l,2 (u), I l,2 (u) and O h,2 (u), I h,2 (u) and O l,h,2 (u), I l,h,2 (u) are the homomorphisms.Thus we deduce that O 2 : K w → OK and O l,2 : K → O l K, I 2 : K w → IK and I l,2 : K → I l K, O h,2 : K w,l → O h K, I h,2 : K w,l → I h K, O l,h,2 : K l → O l,h K and I l,h,2 : K l → I l,h K are the covariant functors on the categories K w , K, K w,l and K l correspondingly with values in the categories of skew idempotent continuous morphisms, when a set X ∈ Ob(S ) or in Ob(S l ) correspondingly is marked.
35.Proposition.The bi-functors I on S × K w , I l on S l × K, I h on S × K w,l and I l,h on S l × K l preserve pre-images.
Proof.In view of Proposition 24 and Lemma 34 I, I l , I h and I l,h are the covariant bi-functors, i.e., the functors in S or S l and the functors in K w or K or K w,l or K l correspondingly as well.For any functor F the inclusion F( f −1 (B)) ⊂ (F( f )) −1 (F(B)) is satisfied, where, for example, B is closed in Y ∈ Ob(S ).
Suppose the contrary that I does not preserve pre-images.This means that there exist X, Y ∈ Ob(S ) and K 1 , There exist functions s, t ∈ C(X, K 1 ) such that (4) s| A = g| A and t| A = h| A , while (5) s| X\A = t| X\A and (6) s(x) ≤ g(x) and s(x) ≤ h(x) for each x ∈ X \ A, where g, h satisfy Conditions (1 − 3) due to property 19(CE).There are also functions q, r ∈ C(X, K 1 ) such that (7) q| X\A = g| X\A and r| X\A = h| X\A with (8) q(x) = r(x) and q(x) ≤ c for each x ∈ A, where On the other hand, there are functions q The condition s = t on A and on X \ A imply that (13) ν(s) = ν(t).Therefore, (14) u(ν(g)) = u(ν(s)) ∨ u(ν(q)) and u(ν(h)) = u(ν(t)) ∨ u(ν(r)), which follows from (10,11).But Formulas (4 − 6, 12 − 14) contradict the inequality u[ν(g)] = u[ν(h)], since u is the order-preserving continuous algebraic homomorphism from K 1 into K 2 .Thus the bi-functors I and I l preserve pre-images.The proof in other cases is analogous.

Corollary
)) correspondingly.37. Definitions.Suppose that Q is a category and F, G are two functors in Q. Suppose also that a transformation p : If T : Q → Q is an endofunctor in a category Q and there are natural transformations the identity η : 1 Q → T and the multiplication ψ : T 2 → T satisfying the relations ψ • Tη = ψ • ηT = 1 T and ψ • ψT = ψ • Tψ, then one says that the triple T := (T, η, ψ) is a monad.
38. Theorem.There are monads in the categories S × K w , S l × K, S × K w,l and S l × K l .
Proof.Let ḡ(ν) := ν(g) for g ∈ C(X, K) and ν ∈ I(X, K), where X ∈ Ob(S ) and K ∈ Ob(K w ).Therefore, this induces the morphism ḡ : I(X, K) → K. Then If additionally ν is left homogeneous and K ∈ Ob(K w,l ), then bg = ν(bg) = bν(g) = b ḡ(ν).Therefore, we infer that bg = b ḡ for every b ∈ K and g ∈ C(X, K). For On the other hand, from Formulas (2, 2 ) we get that Next we verify that the transformations η and ξ are natural for each f ∈ Mor where O m+1 (X, K) := O(O m (X, K), K) for each natural number m (see also §20 and Proposition 35).
3.3.Lattices Associated with Actions of Groupoids on Topological Spaces 40.Lemma.Let G be a topological groupoid with a unit acting on a topological space X such that to each element g ∈ G a continuous mapping v g : X → X corresponds having the properties (1) v g v h = v gh for each g, h ∈ G and (2) v e = id, where e ∈ G is the unit element, id(x) = x for each x ∈ X.If K is a topological ringoid with the associative sub-ringoid L, L ⊃ {0, 1}, such that (3) a(bc) = (ab)c for each a, b ∈ L and c ∈ K, a continuous mapping ρ : G 2 → L \ {0} satisfies the cocycle condition (4) ρ(g, x)ρ(h, v g x) = ρ(gh, x) and (5) ρ(e, x) = 1 ∈ K for each g, h ∈ G and x ∈ X, then (6) T g f (x) := ρ(g, x) vg f (x) is a representation of G by continuous in the g ∈ G variable mappings T g of C(X, K) into C(X, K), when f is marked, where f ∈ C(X, K), vg f (x) := f (v g (x)) for each g ∈ G and x ∈ X.
Proof.For each g, h ∈ G one has T g (T h f (x)) = ρ(g, x) vg [ρ(h, x) vh f (x)] = ρ(gh, x) vgh f (x) = T gh f (x), hence T g T h = T gh .Moreover, T e f = f , since v e = id and ρ(e, x) = 1, i.e., T e = I is the unit operator on C(X, K).Mappings T g f (x) are continuous in the g ∈ G variable as compositions and products of continuous mappings.
The continuous mappings T g are (may be) generally non-linear relative to K. If K is commutative, distributive and associative, then T g are K-linear on C(X, K).
Suppose that G is a topological groupoid with the unit continuously acting on a topological space X and satisfying Conditions 40 (1,2).A continuous morphism λ on C(X, K) or C + (X, K) we call (T, G)-invariant if (2) Tg λ = λ, where ( Tg λ)( f ) := λ(T g f ) for each g ∈ G and f in C(X, K) or C + (X, K) correspondingly.
45. Theorem.If G is a topological groupoid with a unit, X = G as a topological space (see §41), then H j (G, K) is a closed ideal in S j (G, K), where j = + (for K commutative and associative relative to +) or j = ∨ or j = ∧ or j = (∨, h) or j = (∧, h) with ρ(u, x) ≡ 1; j = (∨, h) or j = (∧, h) for commutative and associative K relative to the multiplication with general T u .
Proof.We mention that Tg (b where the operation denoted by the addition + i is either + or ∨ or ∧ for i = 1 or i = 2 or i = 3 correspondingly (and also below in this section), consequently, b 1 λ 1 + i b 2 λ 2 ∈ H j (G, K) for each λ 1 , λ 2 ∈ H j (G, K) and b 1 , b 2 ∈ K, i = i(j).
The property being G-invariant provides closed subsets in S j (G, K), since if a net of continuous mappings g k converges to a continuous mapping g an each g k is G-invariant, then g = lim k g k is G-invariant as well.
and K satisfies Condition(1).5. Remark.For example, the class On of all ordinals has the addition µ 1 = + o and the multiplication µ 2 = × o operations which are generally non-commutative, associative, with unit elements 0 and 1 respectively, on On the right distributivity is satisfied (see Propositions 4.29-4.31and Examples 1-3 in [19]).Relative to the interval topology generated by the base {(a, b) : a < b ∈ On} the class On is the topological well-ordered semiring, where (a, b) = {c : c ∈ On, a < c < b}.
12(3), where a ∨ b = max(a, b) and a ∧ b = min(a, b) for each a, b ∈ K when either a < b or a = b or b < a.

13 .
Definition.If topological spaces X and Y are given and f : X → Y is a continuous mapping, then it induces the mapping O t l = (I l (p b ) : b ∈ V) : I l (X, K) → I l (P, K) and t l,h = (I l,h (p b ) : b ∈ V) : I l,h (X, K) → I l,h (P, K) also are bijective and surjective continuous algebraic homomorphisms.Proof.We consider the inverse system O(P) = (O(X a ); O(p a b ); V} and its limit space Y = lim O(P).Then O(p b a )O(p b ) = O(p a ) for each a ≤ b ∈ V, since p b a the inverse limit decomposition λ = lim{λ b ; O(p b a ); V} and Formula (6) it follows that λ is order-preserving.If X b ∈ Ob(S l ) for each b ∈ V, then a topological space X is linearly ordered: x = {x b : b ∈ V} ≤ y = {y b : b ∈ V} if and only if x b ≤ y b for each b ∈ V, where x, y ∈ X are threads of the inverse system P such that p b a (x b ) = x a for each a ≤ b ∈ V. Since p b a is order-preserving for each a ≤ b ∈ V and each f b is non-decreasing, then f is nondecreasing and hence Thus O := (O, η, ξ) is the monad.Since I is the restriction of the functor O, the triple I := (I, η, ξ) is the monad in the category S × K w as well.Analogously O l := (O l , η, ξ) and I l := (I l , η, ξ) form the monads in the category S l × K; O h = (O h , η, ξ) and I h = (I h , η, ξ) are the monads in S × K w,l ; O l,h = (O l,h , η, ξ) and I l,h = (I l,h , η, ξ) are the monads in S l × K l .39. Proposition.If a sequence