On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations

: This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the deﬁnition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortiﬁed transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being.


Introduction
This paper is generally classified in the area of hypercompositional algebra. Hypercompositional algebra is the branch of abstract algebra which studies hypercompositional structures, i.e., structures equipped with one or more multi-valued operations. Multivalued operations, also called hyperoperations or hypercompositions, are operations in which the result of the synthesis of two elements is multi-valued, rather than a single element. More precisely, a hypercomposition on a non-void set H is a function from H × H to the powerset P(H) of H. Hypercompositional structures came into being through the notion of the hypergroup. The hypergroup was introduced in 1934 by Marty, in order to study problems in non-commutative algebra, such as cosets determined by non-invariant subgroups [1][2][3].
In [4] the magma is defined as an ordered pair (H, ⊥) where H is a non-void set and "⊥" is a law of synthesis, which is either a composition or a hypercomposition. Per this definition, if A and B are subsets of H, then: In particular if A = ∅ or B = ∅, then A⊥B = ∅ and vice versa. A⊥b and a⊥B have the same meaning as A⊥{b} and {a}⊥B. In general, the singleton {a} is identified with its member a. Sometimes it is convenient to use the relational notation A ≈ B instead of A ∩ B = ∅. Then, as the singleton {a} is identified with its member a, the notation a ≈ A or A ≈ a is used as a substitute for a ∈ A or Aa. The relation ≈ may be considered as a weak generalization of equality, since, if A and B are singletons and A ≈ B, then A = B. Thus, a ≈ b⊥c means either a = b⊥c, when the synthesis is a composition or a ∈ b⊥c, when the synthesis is a hypercomposition.
It is possible that the result of the synthesis of a pair of elements in a magma is the void set, when the law of synthesis is a hypercomposition. Then, the structure is called a partial hypergroupoid, otherwise, it is called a hypergroupoid.
Every law of synthesis in a magma induces two new laws of synthesis. If the law of synthesis is written multiplicatively, then the two induced laws are: Thus x ≈ a/b if and only if a ≈ xb and x ≈ b\a if and only if a ≈ bx. In the case of a multiplicative magma, the two induced laws are named inverse laws and they are called the right division and the left division, respectively [4]. It is obvious that, if the law of synthesis is commutative, then the right division and left division coincide.
A law of synthesis on a set H is called associative if the property, (x⊥y)⊥z = x⊥(y⊥z) is valid, for all elements x, y, z in H. A magma whose law of synthesis is associative, is called an associative magma [4].

Definition 1.
An associative magma in which the law of synthesis is a composition is called a semigroup, while it is called a semihypergroup if the law of synthesis is a hypercomposition and ab = ∅ for each pair of its elements.
A law of synthesis (x, y) → x⊥y on a set H is called reproductive if the equality, is valid for all elements x in H. A magma whose law of synthesis is reproductive is called a reproductive magma [4].

Definition 2.
A reproductive magma in which the law of synthesis is a composition is called a quasigroup, while it is called a quasi-hypergroup if the law of synthesis is a hypercomposition and ab = ∅ for each pair of its elements.

Definition 3. [4]
An associative and reproductive magma is called a group, if the law of synthesis is a composition, while it is called a hypergroup if the law of synthesis is a hypercomposition.
The above unified definition of the group and the hypergroup is presented in [4], where it is also proved analytically that it is equivalent to the well-known dominant definition of the group.
A composition or a hypercomposition on a non-void set H is called left inverted associative if: (a⊥b)⊥c = (c⊥b)⊥a, for every a, b, c ∈ H, Mathematics 2021, 9,1828 3 of 31 while it is called right inverted associative if a⊥(b⊥c) = c⊥(b⊥a), for every a, b, c ∈ H.
The notion of the inverted associativity was initially conceived by Kazim and Naseeruddin [5] who endowed a groupoid with the left inverted associativity, thus defining the LA-semigroup. A magma equipped with left inverted assisiativity is called a left inverted associative magma, while if it is equipped with right inverted assisiativity is called a right inverted associative magma.
Recall that if (E, ⊥) is a magma, then the law of synthesis: (x, y) → x⊥ op y = y⊥x is called the opposite of "⊥". The magma (E, ⊥ op ) is called the opposite magma of (E, ⊥) [4]. As it is detailed in [4] the group and the hypergroup satisfy exactly the same axioms, i.e., the reproductive axiom and the associative axiom. This led to the unified definition of the group and the hypergroup that was repeated as Definition 3 in this paper. Using the same approach, the definition of the left/right almost-group and the left/right almost-hypergroup is:

Remark 1.
Obviously if the law of synthesis is commutative, then the LA-or RA-groups and hypergroups are groups and hypergroups respectively, indeed: (a⊥b)⊥c = (c⊥b)⊥a = a⊥(c⊥b) = a⊥(b⊥c) As shown in Theorem 11 in [4], when the law of synthesis is a composition, then the reproductive axiom is valid if and only if the inverse laws are compositions. Hence another definition can be given for the left/right almost-group: Thus, if the law of synthesis on a magma is written multiplicatively, then the magma is a left/right almost-group if and only if it satisfies the axiom of the left/right inverted associativity and both the right and the left division, a/b, b\a respectively, result in a single element, for every pair of elements a, b in the magma.
In a similar way, because of Theorem 14 in [4], if a/b = ∅ and b\a = ∅, for all pairs of elements a, b of a magma, then the magma is reproductive. Therefore, a second definition of the left/right almost-hypergroup can be given:

Definition 6. (SECOND DEFINITION OF THE LEFT/RIGHT ALMOST-HYPERGROUP)
A magma which satisfies the axiom of the left/right inverted associativity is called a left/right almost-hypergroup (LA-hypergroup/RA-hypergroup), if the law of synthesis on the magma is a hypercomposition and the result of each one of the two inverse hypercompositions is nonvoid for all pairs of elements of the magma. Example 1. This example proves the existence of non-trivial left almost-groups (Table 1) and right almost-groups (Table 2). group/RA-group), if the law of synthesis on the magma is a composition, and the two induced laws of synthesis are compositions as well.
Thus, if the law of synthesis on a magma is written multiplicatively, then the magma is a left/right almost-group if and only if it satisfies the axiom of the left/right inverted associativity and both the right and the left division, / a b , \ b a respectively, result in a single element, for every pair of elements , a b in the magma. In a similar way, because of Theorem 14 in [4], if / a b ≠ ∅ and \ b a ≠ ∅ , for all pairs of elements , a b of a magma, then the magma is reproductive. Therefore, a second definition of the left/right almost-hypergroup can be given:   of synthesis are compositions as well.
Thus, if the law of synthesis on a magma is written multiplicatively, then the magma is a left/right almost-group if and only if it satisfies the axiom of the left/right inverted associativity and both the right and the left division, / a b , \ b a respectively, result in a single element, for every pair of elements , a b in the magma. In a similar way, because of Theorem 14 in [4], if / a b ≠ ∅ and \ b a ≠ ∅ , for all pairs of elements , a b of a magma, then the magma is reproductive. Therefore, a second definition of the left/right almost-hypergroup can be given:   (Table 3) and a non-trivial right almost-hypergroup (Table 4).  (Table 3) and a non-trivial right almost-hypergroup (Table 4).      Table 5 which describes a LA-hypergroup is Table 6, which describes a RA-hypergroup.

Remark 2.
It is noted that in the groups, the unified definition which uses only the reproductive axiom and the associative axiom, leads to the existence of a bilaterally neutral element and consequently to the existence of a symmetric element for each one of the group's elements, as it is proved in Theorem 2 in [4]. The same doesn't hold in the case of the left or right almost-groups. Indeed, if e is a neutral element in a left almost-group, then: (a⊥e)⊥c = a⊥c and (a⊥e)⊥c = (c⊥e)⊥a = c⊥a Hence the composition is commutative and therefore the left almost-group is a group.
Special notation: In the following, in addition to the typical algebraic notations, we use Krasner 's notation for the complement and difference. So, we denote with A··B the set of elements that are in the set A, but not in the set B.
Special notation: In the following, in addition to the typical algebraic notations, we use Krasner's notation for the complement and difference. So, we denote with A··B the set of elements that are in the set A, but not in the set B.
A quasicanonical hypergroup or polygroup [26][27][28][29] is a transposition hypergroup H containing a scalar identity, that is, there exists an element e such that ea = ae = a for each a in H. A canonical hypergroup [17][18][19] is a commutative polygroup. A canonical hypergroup may also be characterized as a join hypergroup with a scalar identity [23]. The following Proposition connects the canonical hypergroups with the RA-hypergroups. Proposition 1. Let (H, ·) be a canonical hypergroup and "/" the induced hypercomposition which follows from "·". Then (H, /) is a right almost-hypergroup.
Subsequently, the left and right almost-hypergroups can be enriched with additional axioms. The first axiom to be used for this purpose is the transposition axiom, as it has been introduced into many hypercompositional structures and has given very interesting and useful properties (e.g., see [75,76]).

Definition 7.
If a left almost-hypergroup (H, ·) satisfies the transposition axiom, i.e., b\a ∩ c/d = ∅ implies ad ∩ bc = ∅, f or all a, b, c, d ∈ H then it will be called transposition left almost-hypergroup. If a right almost-hypergroup satisfies the transposition axiom, then it will be called transposition right almost-hypergroup. (Table 7) and a transposition right almost-hypergroup (Table 8).

Example 4. This example presents a transposition left almost-hypergroup
Hence, the right inverted associativity is valued. Moreover, in any hypergroup holds [4,75]: Thus, the reproductive axiom is valid and therefore ( ) Subsequently, the left and right almost-hypergroups can be enriched with additional axioms. The first axiom to be used for this purpose is the transposition axiom, as it has been introduced into many hypercompositional structures and has given very interesting and useful properties (e.g., see [75,76]).

Example 4. This example presents a transposition left almost-hypergroup
In [4] the reverse transposition axiom was introduced: Thus, the following hypercompositional structure can be defined: H ⋅ satisfies the reverse transposition axiom, then it will be called a reverse transposition left (right) almost-hypergroup.
In [4] the reverse transposition axiom was introduced: Thus, the following hypercompositional structure can be defined: Definition 8. If a left (right) almost-hypergroup (H, ·) satisfies the reverse transposition axiom, then it will be called a reverse transposition left (right) almost-hypergroup.

Definition 9. A hypercomposition on a non-void set H is called left inverted weakly associative if
(ab)c ∩ (cb)a = ∅, for every a, b, c ∈ H, while it is called right inverted weakly associative if a(bc) ∩ c(ba) = ∅, for every a, b, c ∈ H.

Definition 10.
A quasi-hypergroup equipped with a hypercomposition which is left inverted weakly associative is called a weak left almost-hypergroup (WLA-hypergroup), while it is called a weak right almost-hypergroup (WRA-hypergroup) if the hypercomposition is right inverted weakly associative.
Recall that a quasi-hypergroup which satisfies the weak associativity is called H Vgroup [77].
Proof. Suppose that H is a commutative WLA-hypergroup, then: Hence H is an H V -group. Proposition 3. Let (H, ·) be a left almost-hypergroup. An arbitrary subset I ab of H is associated to each pair of elements (a, b) ∈ H 2 and the following hypercomposition is introduced into H: Then (H, * ) is a WLA-hypergroup.
Proof. Since xH and Hx are subsets of x * H and H * x respectively, it follows that the reproductive axiom holds. On the other hand: It is obvious that if the composition or the hypercomposition is commutative, then the inverted associativity coincides with the associativity. Thus, a commutative LA-hypergroup is simply a commutative hypergroup. However, in the hypercompositions there exists a property that does not appear in the compositions. This is the weak commutativity. A hypercomposition on a non-void set H is called weakly commutative if ab ∩ ba = ∅, for all a, b ∈ H. Definition 11. A left almost commutative hypergroup (LAC-hypergroup) is a left almost-hypergroup which satisfies the weak commutativity. A right almost commutative hypergroup (RAC-hypergroup) is a right almost-hypergroup which satisfies the weak commutativity. A LAC-hypergroup (resp. RAC-hypergroup) which satisfies the transposition axiom will be called join left almost-hypergroup (resp. reverse join left almost-hypergroup). A LAC-hypergroup (resp. RAC-hypergroup) which satisfies the reverse transposition axiom will be called reverse join left almost-hypergroup (resp. reverse join right almost-hypergroup). Example 5. This example presents a join left almost-hypergroup (Table 9) and a join right almosthypergroup (Table 10). Table 9. Join left almost-hypergroup.
It is obvious that if the composition or the hypercomposition is commutative, then the inverted associativity coincides with the associativity. Thus, a commutative LAhypergroup is simply a commutative hypergroup. However, in hypercompositions there exists a property that does not appear in the compositions. This is the weak commutativity.  (Table 9) and a join right almosthypergroup (Table 10). Table 9. Join left almost-hypergroup.
A weak left almost-hypergroup which satisfies the weak commutativity will be named a weak left almost commutative hypergroup (WLAC-hypergroup). Analogous is the definition of the weak right almost commutative hypergroup (WRAC-hypergroup), i. ii.

H * is a weak join left almost-hypergroup
A weak left almost-hypergroup which satisfies the weak commutativity will be named a weak left almost commutative hypergroup (WLAC-hypergroup). Analogous is the definition of the weak right almost commutative hypergroup (WRAC-hypergroup),   Remark 3. Analogous propositions to the above 3-7, hold for the right almost-hypergroups as well.

Enumeration and Structure Results
The enumeration of hypercompositional structures is the subject of several papers (e.g., [78][79][80][81][82][83][84][85][86][87]). In [78] a symbolic manipulation package is developed which enumerates the hypergroups of order 3, separates them into isomorphism classes and calculates their cardinality. Following analogous techniques, in this paper, a package is developed (see Appendix A) which, when combined with the package in [78], enumerates the left almost-hypergroups and the right almost-hypergroups with 3 elements, classifies them in isomorphism classes and computes their cardinality.
For the purpose of the package, the set H = {1, 2, 3} is used as the set with three elements. The laws of synthesis in H are defined through the Cayley Table 11:  where the intersection of row i with column j, i.e., aij, is the result of i•j. Αs in the case of hypergroups, in the left almost-hypergroups or right almost-hypergroups, the result of the hypercomposition of any two elements is non-void (see Theorem 4 below). Thus, the cardinality of the set of all possible magmas with 3 elements which are not partial hypergroupoids, is 7 9 = 40 353 607. As it is mentioned above in the commutative case the left inverted associativity and the right inverted associativity coincide with the associativity. Among the 40 353 607 magmas only 2 520 are commutative hypergroups, which are the trivial cases of left almost-hypergroups and right almost-hypergroups. Thus, the package focuses on the non-trivial cases, that is on the non-commutative magmas. The enumeration reveals that there exist 65 955 reproductive non-commutative magmas which satisfy the left inverted associativity only and obviously the same number of reproductive non-commutative magmas which satisfy the right inverted associativity only. That is, there exist 65 955 non-trivial left almost-hypergroups and 65 955 non-trivial right almost-hypergroups. Moreover, there exist 7 036 reproductive magmas which satisfy both the left and the right inverted associativity i.e., there exist 7 036 both left and right almost-hypergroups. Furthermore, there are 16 044 reproductive magmas that satisfy the left inverted associativity, the right inverted associativity and the associativity. This means that there exist 16 044 structures which are simultaneously left almosthypergroups, right almost-hypergroups and hypergroups. Finally, there do not exist any reproductive magmas which satisfies both the left (or right) inverted associativity and the associativity.
The following examples present the worth mentioning cases in which a hypercompositional structure is (a) both left and right almost-hypergroup and (b) simultaneously left almost-hypergroup, right almost-hypergroup and non-commutative hypergroup.
Example 6. The hypercompositional structure described in Cayley Table 12 is both left and right almost-hypergroup.
where the intersection of row i with column j, i.e., a ij, is the result of i•j. As in the case of hypergroups, in the left almost-hypergroups or right almost-hypergroups, the result of the hypercomposition of any two elements is non-void (see Theorem 4 below). Thus, the cardinality of the set of all possible magmas with 3 elements which are not partial hypergroupoids, is 7 9 = 40 353 607. As it is mentioned above in the commutative case the left inverted associativity and the right inverted associativity coincide with the associativity. Among the 40 353 607 magmas only 2520 are commutative hypergroups, which are the trivial cases of left almost-hypergroups and right almost-hypergroups. Thus, the package focuses on the non-trivial cases, that is on the non-commutative magmas. The enumeration reveals that there exist 65 955 reproductive non-commutative magmas which satisfy the left inverted associativity only and obviously the same number of reproductive noncommutative magmas which satisfy the right inverted associativity only. That is, there exist 65 955 non-trivial left almost-hypergroups and 65 955 non-trivial right almost-hypergroups. Moreover, there exist 7036 reproductive magmas which satisfy both the left and the right inverted associativity i.e., there exist 7036 both left and right almost-hypergroups. Furthermore, there are 16 044 reproductive magmas that satisfy the left inverted associativity, the right inverted associativity and the associativity. This means that there exist 16 044 structures which are simultaneously left almost-hypergroups, right almost-hypergroups and hypergroups. Finally, there do not exist any reproductive magmas which satisfies both the left (or right) inverted associativity and the associativity.
The following examples present the worth mentioning cases in which a hypercompositional structure is (a) both left and right almost-hypergroup and (b) simultaneously left almost-hypergroup, right almost-hypergroup and non-commutative hypergroup. Example 6. The hypercompositional structure described in Cayley Table 12 is both left and right almost-hypergroup. hypergroups, right almost-hypergroups and hypergroups. Finally, there do not exist any reproductive magmas which satisfies both the left (or right) inverted associativity and the associativity.
The following examples present the worth mentioning cases in which a hypercompositional structure is (a) both left and right almost-hypergroup and (b) simultaneously left almost-hypergroup, right almost-hypergroup and non-commutative hypergroup. Table 12 is both left and right almost-hypergroup.   A magma though, with three elements, can be isomorphic to another magma, which results from a permutation of these three elements. The isomorphic structures which appear in this way, can be considered as members of the same class. These classes can be constructed and enumerated, with the use of the techniques and methods which are developed in [78]. Having done so, the following conclusions occurred: A magma though, with three elements, can be isomorphic to another magma, which results from a permutation of these three elements. The isomorphic structures which appear in this way, can be considered as members of the same class. These classes can be constructed and enumerated, with the use of the techniques and methods which are developed in [78]. Having done so, the following conclusions occurred: The above results, combined with the ones of [78] for the hypergroups, are summarized and presented in the following Table 14: In the above table we observe that there is only one class of left almost-hypergroups and only one class of right almost-hypergroups which contains a single member. The member of this class is of particular interest, since its automorphism group is of order 1. Such hypercompositional structures are called rigid. Additionally, observe that there are 6 rigid hypergroups all of which are commutative. A study and enumeration of these rigid hypergroups, as well as other rigid hypercompositional structures is given in [81]. The following Table 15 presents the unique rigid left almost-hypergroup, while Table 16 describes the unique rigid right almost-hypergroup with 3 elements. In the above table we observe that there is only one class of left almost-hypergroups and only one class of right almost-hypergroups which contains a single member. The member of this class is of particular interest, since its automorphism group is of order 1. Such hypercompositional structures are called rigid. Additionally, observe that there are 6 rigid hypergroups all of which are commutative. Α study and enumeration of these rigid hypergroups, as well as other rigid hypercompositional structures is given in [81]. The following Table 15 presents the unique rigid left almost-hypergroup, while Table 16 describes the unique rigid right almost-hypergroup with 3 elements.       [82].

Example 6. The hypercompositional structure described in Cayley
More generally, the next theorem is valid:

Algebraic Properties
In hypercompositional algebra it is dominant that a hypercomposition on a set E is a mapping of E × E to the non-void subsets of E. In [4], it is shown that this restriction is not necessary, since it can be proved that the result of the hypercomposition is non void in many hypercompositional structures. Such is the hypergroup (see [4], Theorem 12,and [75], Property 1.1), the weakly associative magma ( [4], Proposition 5) and consequently the H V -group ( [76], Proposition 3.1). The next theorem shows that in the case of the left/right almost hypergroups the result of the hypercomposition is non-void as well. The proof is similar to the one in [4], Theorem 12, but not the same due to the validity of the inverted associativity in these structures, instead of the associativity.

Proposition 8. A left almost-hypergroup H is a hypergroup if and only if a(bc) = (cb)a holds for all a, b, c in H.
Proof. Let H be a hypergroup. Then, the associativity (ab)c = a(bc) holds for all a, b, c in H. Moreover because of the assumption, a(bc) = (cb)a is valid for all a, b, c in H. Therefore (ab)c = (cb)a, thus H is a left almost-hypergroup. Conversely now, suppose that a(bc) = (cb)a holds for all a, b, c in H. Since H is a left almost-hypergroup, the sequence of the equalities a(bc) = (cb)a = (ab)c is valid. Consequently, H is a hypergroup. Proof. (i) Because of the reproduction, Hb = H for all b ∈ H. Hence, for every a ∈ H there exists x ∈ H, such that a ∈ xb. Thus, x ∈ a/b and, therefore, a/b = ∅. Similarly, b\a = ∅.
(ii) Because of Theorem 4, the result of the hypercomposition in H is always a nonempty set. Thus, for every x ∈ H there exists y ∈ H, such that y ∈ xa, which implies that x ∈ y/a. Hence, H ⊆ H/a. Moreover, H/a ⊆ H. Therefore, H = H/a. Next, let x ∈ H. Since H = xH, there exists y ∈ H such that a ∈ xy, which implies that x ∈ a/y. Hence, H ⊆ a/H. Moreover, a/H ⊆ H. Therefore, H = a/H.
(iii) Suppose that x/a = ∅, for all a, x ∈ H. Thus, there exists y ∈ H, such that x ∈ ya. Therefore, x ∈ Ha, for all x ∈ H, and so H ⊆ Ha. Next, since Ha ⊆ H for all a ∈ H, it follows that H = Ha. Per duality, H = aH. Conversely now, per Theorem 4, the reproductivity implies that a/b = ∅ and b\a = ∅, for all a, b in H. (c\b)a ∪ (b\c)\a ⊆ c\(ba) for all a, b, c in H. (1) and (2) it follows that xc ∩ ab = ∅. So, there exists z ∈ ab, such that z ∈ xc which implies that x ∈ z/c. Hence, x ∈ (ab)/c. Thus (i) is valid. Similar is the proof of (ii). Proof. (i) Let x ∈ (b\a)(c/d). Then, because of Proposition 9.i, there exists y ∈ b\a, such that x ∈ y(c/d) ⊆ (yc)/d. Thus xd ∩ yc = ∅ or xd ∩ (b\a)c = ∅. Because of Proposition 9.ii, it holds that: (b\a)c ⊆ b\ac. Therefore x ∈ (b\ac)/d (1). Next, since x ∈ (b\a)(c/d), there exists z ∈ c/d, such that x ∈ (b\a)z. Because of Proposition 9.ii, the inclusion relation (b\a)z ⊆ b\(a z) holds. Thus, bx ∩ az = ∅ or equivalently bx ∩ a(c/d) = ∅ or, because of Proposition 9.i, bx ∩ ac/d = ∅. Therefore, x ∈ b\(ac/d) (2). Now (1) and (2) give (i).
(iii) can be proved in a similar manner. b ∈ a/(b\a).
Proof. (i) Let x ∈ a/b. Then a ∈ xb. Hence, b ∈ x\a. Thus, b ∈ (a/b)\a. Therefore, (i) is valid. The proof of (ii) is similar. Remark 5. The above properties are consequences of the reproductive axiom and therefore are valid in both, the left and the right almost-hypergroups as well as in the hypergroups [39].
In any left almost-hypergroup the following property is valid: (a/b)/c = (bc)\a (mixed left inverted associativity) ii.
In any right almost-hypergroup the following property is valid: c\(b\a) = a/(cb) (mixed right inverted associativity).
Proof. (i) Let x ∈ (a/b)/c. Then the following sequence of equivalent statements is valid: Similar is the proof of (ii). i.
In any left almost-hypergroup the right inverted associativity of the induced hypercompositions is valid: b\(a/c) = c\(a/b). ii.
In any right almost-hypergroup the left inverted associativity of the induced hypercompositions is valid: (b\a)/c = (c\a)/b.
Proof. For (i) it holds that: Regarding (ii) it is true that: This completes the proof.

Corollary 9.
i. If A, B, C are non-empty subsets of a left almost-hypergroup H, then: ii. If A, B, C are non-empty subsets of a right almost-hypergroup H, then: (B\A)/C = (C\A)/B.

Identities and Symmetric Elements
Let H be a non-void set endowed with a hypercomposition. An element e of H is called right identity, if x ∈ x · e for all x in H. If x ∈ e · x for all x in H, then e is called left identity, while e is called identity if it is both right and left identity. An element e of H is called right scalar identity, if x = x · e for all x in H. If x = e · x for all x in H, then e is called left scalar identity, while e is called scalar identity if it is both right and left scalar identity [4,13,18,88]. When a left (resp. right) scalar identity exists in H, then it is unique. Tables 17 and 18 describe left/right almost-hypergroups with left/right scalar identity.  a c c a b ii.

Example 8. The Cayley
In any right almost-hypergroup the left inverted associativity of the induced hypercompositions is valid: a c c a b Proof. For (i) it holds that: x H a cx c a b = ∈ ∈ = ∈ ∩ ≠∅ = Regarding (ii) it is true that: This completes the proof. □ Corollary 9.

i. If
, , A B C are non-empty subsets of a left almost-hypergroup H , then: ii. If

Identities and Symmetric Elements
Let H be a non-void set endowed with a hypercomposition. An element e of H is called right identity, if ∈ ⋅ x x e for all x in .     [4,32,34,39,[88][89][90]. Note that the strong identity needs not be unique [32,34,39]. Both the scalar identity and the strong identity are idempotent identities. If the equality e = ee is valid for an identity e, then e is called idempotent identity [4,39,88]. e is a right strong identity, if x ∈ x · e ⊆ {e, x} for all x in H while e is a left strong identity, if x ∈ e · x ⊆ {e, x}. e is a strong identity, if it is right and left strong [4,32,34,39,[88][89][90]. Note that the strong identity needs not be unique [32,34,39]. Both the scalar identity and the strong identity are idempotent identities.

Theorem 6. If e is a strong identity in a left almost-hypergroup H, then
x/e = e\x = x, for all x ∈ H − {e} Proof. Suppose that y ∈ x/e. Then x ∈ ye ⊆ {y, e}. Consequently y = x.
Let e be an identity element in H and x an element in H. Then, x will be called right e-attractive, if e ∈ e · x, while it will be called left e-attractive if e ∈ x · e. If x is both left and right e-attractive, then it will be called e-attractive. When there is no likelihood of confusion, e can be omitted. See [32] for the origin of the terminology. i.
x ∈ x/y and x ∈ y\x, for all x, y ∈ H ii.
x/x = x\x = H, for all x, y ∈ H Proposition 20. If a left almost-hypergroup or a right almost-hypergroup H has a strong identity e and if the relation e ∈ bc implies that e ∈ cb, for all b, c ∈ H, then H is a hypergroup. According to the assumption if e ∈ bc, then e ∈ cb. Hence bc ⊆ cb. Thus cb = bc. Next, if e / ∈ bc then e / ∈ cb and (i) implies that cb ⊆ bc while (ii) implies that bc ⊆ cb. Thus cb = bc. Consequently H is commutative and therefore H is a hypergroup.

Proposition 21. If a left almost-hypergroup or a right almost-hypergroup H has a strong identity e and if A is the set of its attractive elements, then
A/A ⊆ A and A\A ⊆ A.
Proof. Since A is the set of the attractive elements, e/e = A and e\e = A is valid. Therefore: An element x is called right e-inverse or right e-symmetric of x, if there exists a right identity x = e such that e ∈ x · x . The definition of the left e-inverse or left e-symmetric is analogous to the above, while x is called e-inverse or e-symmetric of x, if it is both right and left inverse with regard to the same identity e. If e is an identity in a left almost-hypergroup H, then the set of the left inverses of x ∈ H, with regard to e, will be denoted by S el (x), while S er (x) will denote the set of the right inverses of x ∈ H with regard to e. The intersection S el (x) ∩ S er (x) will be denoted by S e (x).
Proof. y ∈ e/x, if and only if e ∈ yx. This means that either y ∈ S el (x) or y = e, if x is right attractive. Hence, e/x ⊆ {e} ∪ S el (x). The rest follows in a similar way. ey ∩ x z = ∅, for all x ∈ S el (x), ii.
Proof. z ∈ xy implies that x ∈ z/y and that y ∈ x\z. Let x ∈ S el (x) and y ∈ S er (y). Then e ∈ x x and e ∈ yy . Thus x ∈ x \e and y ∈ e\y . Therefore x \e ∩ z/y = ∅ and x\z ∩ e/y = ∅. Hence, because of the transposition, ey ∩ x z = ∅ and xe ∩ zy = ∅.

Proposition 24.
Let H be a transposition left almost-hypergroup with a strong identity e and let x, y, z be elements in H such that x, y, z = e and z ∈ xy. Then: if S er (y) ∩ S er (z) = ∅, then x ∈ zy , for all y ∈ S er (y).
Proof. (i) According to Proposition 23, z ∈ xy implies that ey ∩ x z = ∅ for all x ∈ S l (x). Since e is strong ey = {e, y}. Hence {e, y} ∩ x z = ∅. But S el (x) and S el (z) are disjoint. Thus e / ∈ x z, therefore y ∈ x z. Analogous is the proof of (ii).

Substructures of the Left/Right Almost-Hypergroups
There is a big variety of substructures in the hypergroups, which is much bigger than the one in the groups. Analogous is the variety of the substructures which are revealed here in the case of the left/right almost-hypergroups. For the consistency of the terminology [4,13,33,55,[91][92][93][94][95][96][97][98][99], the terms semisub-left/right almost-hypergroup, sub-left/right almost-hypergroup, etc. will be used in exactly the same way as the prefixes sub-and semisub-are used in the cases of the groups and the hypergroups, e.g., the terms subgroup, subhypergroup are used instead of hypersubgroup, etc. The following research is inspired by the methods and techniques used in [93][94][95][96][97].
Let H be a left almost-hypergroup. Then,

Substructures of the Left/Right Almost-Hypergroups
There is a big variety of substructures in the hypergroups, which is much bigger than the one in the groups. Analogous is the variety of the substructures which are revealed here in the case of the left/right almost-hypergroups. For the consistency of the terminology [4,13,33,55,[91][92][93][94][95][96][97][98][99], the terms semisub-left/right almost-hypergroup, subleft/right almost-hypergroup, etc. will be used in exactly the same way as the prefixes suband semisub-are used in the cases of the groups and the hypergroups, e.g., the terms subgroup, subhypergroup are used instead of hypersubgroup, etc. The following research is inspired by the methods and techniques used in [93][94][95][96][97].
Let H be a left almost-hypergroup. Then,

Fortification in Transposition Left Almost-Hypergroups
The transposition left almost-hypergroups can be fortified through the introduction of neutral elements. Next, we will present two such hypercompositional structures. e is a left identity and ee = e ii.
for every x ∈ H · ·{e} there exists a unique y ∈ H · ·{e} such that e ∈ xy and e ∈ yx For x ∈ H · ·{e} the notation x −1 is used for the unique element of H · ·{e} that satisfies axiom (ii). Clearly x −1 −1 = x. The next results are obvious.

Proposition 38.
i If x = e, then e\x = x. ii e\e = H.
Proposition 39. Let x ∈ H · ·{e}. Then e ∈ xy or e ∈ yx implies y ∈ x −1 , e . Now the role of the identity in the transposition left almost-hypergroups can be clarified.

Proposition 40.
The identity e of the transposition left almost-hypergroup is left strong.
Proof. It suffices to prove that ex ⊆ {e, x}. For x = e the inclusion is valid. Let x = e. Suppose that y ∈ ex. Then x ∈ e\y. But e ∈ xx −1 , hence x ∈ e/x −1 . Thus, (e\y) ∩ e/x −1 = ∅. The transposition axiom gives e = ee = yx −1 . By the previous proposition, y ∈ {e, x}. Therefore, the proposition holds.

Proposition 41.
The identity e of the transposition left almost-hypergroup is unique.
Proof. Suppose that u is an identity distinct from e. Then, there would exist the inverse of e, i.e., an element v distinct from u such that u ∈ ev, which is absurd because ev ⊆ {e, v}. for every x ∈ P · ·{e} there exists at least one element x ∈ P · ·{e}, a symmetric of x , such that e ∈ x x which furthermore satisfies e ∈ x x.
The set of the symmetric elements of x is denoted by S(x) and it is called the symmetric set of x. Table 20 describes a transposition polysymmetrical left almost-hypergroup in which, the element 1 is a left idempotent identity.   Proof. Let y = e and y ∈ ex, then x ∈ e\y. Moreover, for every x ∈ S(x) it holds x ∈ e/x . Consequently e/x ∩ e\y = ∅, so, per transposition axiom, ee ∩ yx = ∅, that is e ∈ yx and thus y ∈ S(x ) ⊆ S(S(x)). Proposition 46. If x = e, is a right attractive element of a transposition polysymmetrical left almost-hypergroup, then S(x) consists of left attractive elements.

Example 10. Cayley
Proof. Let e ∈ ex. Then x ∈ e\e. Moreover, if x is an arbitrary element from S(x), then e ∈ xx . Therefore x ∈ e/x . Consequently e\e ∩ e/x = ∅. Per transposition axiom ee ∩ x e = ∅ . So e ∈ x e. Thus x is an attractive element. ii. If x is a left attractive element of a transposition polysymmetrical left almost-hypergroup, then all the elements of ex are left attractive.

Proof.
Assuming that x is a right attractive element we have that e ∈ ex, which implies that x ∈ e\e. Additionally, if z is an element in xe, then x ∈ z/e. Thus (z/e) ∩ (e\e) = ∅. Per transposition axiom, (ee) ∩ (ez) = ∅ and therefore e ∈ ez, i.e., z is right attractive. Similar is the proof of (ii).

Conclusions and Open Problems
In [70,93] and later in [4], with more details, it was proved that the group can be defined with the use of two axioms only: the associativity and the reproductivity. Likewise, the left/right almost-group is defined here and their existence is proved via examples. The study of this structure reveals a very interesting research area in abstract algebra.
This paper focuses on the study of the more general structure, i.e., of the left/right almost-hypergroup. The enumeration of these structures showed that they appear more frequently than the hypergroups. Indeed, in the case of the hypergroupoids with three elements, there exists one hypergroup in every 1740 hypergroupoids, while there is one non-trivial purely left almost-hypergroup in every 612 hypergroupoids. The same holds for the non-trivial purely right almost-hypergroups, as it is proved in this paper that the cardinal number of the left almost-hypergroups is equal to the cardinal number of the right almost-hypergroups, over a set E. Moreover, there is one non-trivial left and right almost-hypergroup in every 5735 hypergroupoids. Considering the trivial cases as well, i.e., left and right almost-hypergroups which are also non-commutative hypergroups, there exists one left almost-hypergroup in every 453 hypergroupoids. This frequency, which is nearly 4 times higher than that of the hypergroups, justifies a more thorough study of these structures.
Subsequently, these structures were equipped with more axioms, the first one of which is the transposition axiom: b\a ∩ c/d = ∅ implies ad ∩ bc = ∅, for all a, b, c, d ∈ H The transposition left/right almost-hypergroup is studied here.
In [4] though, the reverse transposition axiom was introduced: ad ∩ bc = ∅ implies b\a ∩ c/d = ∅, for all a, b, c, d ∈ H The study of the reverse transposition left/right almost-hypergroup is an open problem. Additionally, open problems for algebraic research are the studies of the properties of all the structures which are introduced in this paper (weak left/right almost-hypergroup, left/right almost commutative hypergroup, join left/right almost hypergroup, reverse join left/right almost hypergroup, weak left/right almost commutative hypergroup) as well as their enumerations. Especially, for the enumeration problems, it is worth mentioning those, which are associated with the rigid hypercompositional structures, that is hypercompositional structures whose automorphism group is of order 1. The conjecture is that there exists only one rigid left almost-hypergroup (and one rigid right almost-hypergroup), while it is known that there exist six such hypergroups, five of which are transposition hypergroups [81].

Conflicts of Interest:
The authors declare no conflict of interest.