Generally speaking, the embedding of an algebraic structure into another one requires the existence of an injective map between the two algebraic objects, that also preserves the structure, i.e., a monomorphism. The most natural, canonical and well-known embeddings are those of numbers: the natural numbers into integers, the integers into the rational numbers, the rational numbers into the real numbers and the real numbers into the complex numbers. One important type of rings is that one of the endomorphisms of an abelian group under function pointwise addition and composition of functions. It is well known that every ring is isomorphic with a subring of such a ring of endomorphisms. But this result holds only in the commutative case, since the set of the endomorphisms of a non-abelian group is no longer closed under addition. This aspect motivates the interest in studying near-rings, that appear to have applications also in characterizing transformations of a group. More exactly, the set of all transformations of a group G, i.e., can be endowed with a near-ring structure under pointwise addition and composition of mappings, such a near-ring being called the transformation near-ring of the group G.
In 1959 Berman and Silverman [1
] claimed that every near-ring is isomorphic with a near-ring of transformations. At that time only some hints were presented, while a direct and clear proof of this result appeared in Malone and Heatherly [2
] almost ten years later. Since
has an identity, it immediately follows that any near-ring can be embedded in a near-ring with identity. Moreover, in the same paper [2
], it was proved that a group
can be embedded in a group
if and only if the near-ring
, consisting of all transformations of H
which multiplicatively commute with the zero transformation, can be embedded into the similar near-ring
under a kernel-preserving monomorphism of near-rings.
Similarly to near-rings, but in the framework of algebraic hyperstructures, Dašić [3
] defined the hypernear-rings as hyperstructures with the additive part being a quasicanonical hypergroup [4
] (called also a polygroup [6
]), and the multiplicative part being a semigroup with a bilaterally absorbing element, such that the multiplication is distributive with respect to the hyperaddition on the left-hand side. Later on, this algebraic hyperstructure was called a strongly distributive hypernear-ring
, or a zero-symmetric hypernear-ring
, while in a hypernear-ring the distributivity property was replaced by the “inclusive distributivity” from the left (or right) side. Moreover, when the additive part is a hypergroup and all the other properties related to the multiplication are conserved, we talk about a general hypernear-ring
]. The distributivity property is important also in other types of hyperstructures, see e.g., [9
]. A detailed discussion about the terminology related to hypernear-rings is included in [10
]. In the same paper, the authors defined on the set of all transformations of a quasicanonical hypergroup that preserves the zero element a hyperaddition and a multiplication (as the composition of functions) in such a way to obtain a hypernear-ring. More general, the set of all transformations of a hypergroup (not necessarily commutative) together with the same hyperaddition and multiplication is a strongly distributive hypernear-ring [3
]. In this note we will extend the study to the set of all multimappings (or multitransformations) of a (non-abelian) hypergroup, defining first a structure of (left) general hypernear-ring, called the multitransformation general hypernear-ring associated with a hypergroup. Then we will show that any hypernear-ring can be weakly embedded into a multitransformation general hypernear-ring, generalizing the similar classical theorem on near-rings [2
]. Besides, under same conditions, any additive hypernear-ring is weakly embeddable into the additive hypernear-ring of the transformations of a hypergroup with identity element that commute multiplicatively with the zero-function. The paper ends with some conclusive ideas and suggestions of future works on this topic.
We start with some basic definitions and results in the framework of hypernear-rings and near-rings of group mappings. For further properties of these concepts we refer the reader to the papers [2
] and the fundamental books [13
]. For the consistence of our study, regarding hypernear-rings we keep the terminology established and explained in [8
First we recall the definition introduced by Dašić in 1978.
Definition 1.  A hypernear-ring is an algebraic system , where R is a non-empty set endowed with a hyperoperation and an operation , satisfying the following three axioms:
is a quasicanonical hypergroup (named also polygroup ), meaning that:
for any ,
there exists such that, for any ,
for any there exists a unique element such that ,
for any implies that .
is a semigroup endowed with a two-sided absorbing element 0, i.e., for any
The operation “·” is distributive with respect to the hyperoperation “+” from the left-hand side: for any , there is
This kind of hypernear-ring was called by Gontineac [11
] a zero-symmetric hypernear-ring
. In our previous works [10
], regarding the distributivity, we kept the Vougiouklis’ terminology [17
], and therefore, we say that a hypernear-ring is a hyperstructure
satisfying the above mentioned axioms
, and the new one:
. The operation “·” is inclusively distributive with respect to the hyperoperation “+” from the left-hand side: for any . Accordingly, the Dašić ’s hypernear-ring (satisfying the axioms , , and ) is called strongly distributive hypernear-ring.
Furthermore, if the additive part is a hypergroup (and not a polygroup), then we talk about a more general type of hypernear-rings.
Definition 2.  A general (left) hypernear-ring is an algebraic structure such that is a hypergroup, is a semihypergroup and the hyperoperation “·” is inclusively distributive with respect to the hyperoperation “+” from the left-hand side, i.e., , for any . If in the third condition the equality is valid, then the structure is called strongly distributive general (left) hypernear-ring. Besides, if the multiplicative part is only a semigroup (instead of a semihypergroup), we get the notion of general (left) additive hypernear-ring. Definition 3.
Let and be two general hypernear-rings. A map is called an inclusion homomorphism if the following conditions are satisfied:
for all .
A map ρ is called a good (strong) homomorphism if in the conditions and the equality is valid.
In the second part of this section we will briefly recall the fundamentals on near-rings of group mappings. A left near-ring is a non-empty set endowed with two binary operations, the addition + and the multiplication ·, such that is a group (not necessarily abelian) with the neutral element 0, is a semigroup, and the multiplication is distributive with respect to the addition from the left-hand side. Similarly, we have a right near-ring. Several examples of near-rings are obtained on the set of “non-linear” mappings and here we will see two of them.
be a group (not necessarily commutative) and let
be the set of all functions from G
define two binary operations: “+” is the pointwise addition of functions, while the multiplication “·” is the composition of functions. Then
is a (left) near-ring, called the transformation near-ring
on the group G
. Moreover, let
be the subnear-ring of
consisting of the functions of
that commute multiplicatively with the zero function, i.e.,
. These two near-rings,
, have a fundamental role in embeddings. Already in 1959, it was claimed by Berman and Silverman [1
] that every near-ring is isomorphic with a near-ring of transformations. One year later the proof was given by the same authors, but using an elaborate terminology and methodology. Here below we recall this result together with other related properties, as presented by Malone and Heatherly [2
Theorem 1.  Let be a near-ring. If is any group containing as a proper subgroup, then can be embedded in the transformation near-ring . Corollary 1.  Every near-ring can be embedded in a near-ring with identity. Theorem 2.  A group can be embedded in a group if and only if can be embedded in by a near-ring monomorphism which is kernel-preserving. Theorem 3.  A group can be embedded in a group if and only if the near-ring can be embedded in the near-ring .
3. Weak Embeddable Hypernear-Rings
In this section we aim to extend the results related to embeddings of near-rings to the case of hypernear-rings. In this respect, instead of a group we will consider a hypergroup and then the set of all multimappings on H, which we endow with a structure of general hypernear-ring.
Theorem 4. Let be a hypergroup (not necessarily abelian) and the set of all multimappings of the hypergroup . Define, for all , the following hyperoperations:
The structure is a (left) general hypernear-ring.
. Indeed, for any
, it holds
. Therefore, for the map
, it holds
. Now, we prove that the hyperoperation ⊕ is associative. Let
Thus, if , then, for all , it holds: . Conversely, if is an element of such that: for all , and if we choose such that for all then and i.e., So, . On the other side, take . Then, By the associativity of the hyperoperation + we obtain that , meaning that the hyperoperation ⊕ is associative.
Let . We prove that the equation has a solution . If we set , for all , then and for all it holds . So, . Similarly, the equation has a solution in . Thus, is a hypergroup.
Now, we show that
is a semihypergroup. Let
. For all
be a multimapping defined by
, for all
. Let us prove that ⊙ is a associative. Let
So, if , then , for all . On the other side, if and for all , then we choose such that and consequently we obtain that . Thus, . So, .
Similarly, . Thus, .
It remains to prove that the hyperoperation ⊕ is inclusively distributive with respect to the hyperoperation ⊙ on the left-hand side. Let Set . So, if then for all it holds: .
On the other hand, Let . Choose, such that and for all . Then and . Thus, for all , i.e., and . So, . Therefore, . □
is called the multitransformations general hypernear-ring on the hypergroup H.
Let be a group and be the transformations near-ring on G. Obviously, and, for all , it holds: , , meaning that the hyperoperations defined in Theorem 4 are the same as the operations in Theorem 1. It follows that is a sub(hyper)near-ring of .
We say that the hypernear-ring is weak embeddable (by short embeddable) in the hypernear-ring if there exists an injective inclusion homomorphism .
The next theorem is a generalization of Theorem 1 [5
For every general hypernear-ring there exists a hypergroup such that R is embeddable in the associated hypernear-ring .
be a hypernear-ring and let
be a hypergroup such that
is a proper subhypergroup of
. For a fixed element
we define a multimapping
Let us define now the mapping as , which is an inclusion homomorphism. Indeed, if then we have and
Consider and If , then . If , then . It follows that, for all , we have and therefore , meaning that .
Similarly, there is and . Let . Then, for , it holds: If , then there is . Thus, and so .
Based on Definition 3, we conclude that is an inclusive homomorphism. It remains to show that is injective. If , then for all , it holds . So, if we choose , then we get that .
These all show that the general hypernear-ring R is W-embeddable in . □
If is a near-ring such that is a proper subgroup of a group , then for a fixed the multimapping constructed in the proof of Theorem 5 is in fact a map from G to G, since in this case the multiplication · is an ordinary operation, i.e., , for all . Thus and thereby . By consequence is an ordinary monomorphism. In other words, Theorem 5 is a generalization of Theorem 1.
Example 1. Let be a left near-ring. Let and be non-empty subsets of R such that and , where is the center of R, i.e., . For any define:
Then the structure is a general left hypernear-ring [8,18]. Let and define on H the hyperoperation as follows:
It is clear that H is a hypergroup such that is a proper subhypergroup of . Besides, based on Theorem 5, for every the multimapping is defined as
Clearly it follows that , defined by , is an inclusive homomorphism, so the general left hypernear-ring is W-embeddable in .
Example 2. Consider the semigroup of natural numbers with the standard multiplication operation and the order “≤”. Define on it the hyperoperations and as follows:
Then the structure is a strongly distributive general hypernear-ring (in fact it is a hyperring). This follows from Theorem . Furthermore, for any , it can be easily verified that is a proper subhypergroup of , where the hyperoperation is defined by:
In this case, for a fixed , we can define the multimapping as follows:
and therefore the mapping is an inclusive homomorphism. Again this shows that the general hypernear-ring is W-embeddable in .
Example 3. Let . Consider now the semigroup defined by Table 1: Define on R the hyperoperation as follows: , so its Cayley table is described in Table 2: Obviously, the relation ≤ is reflexive and transitive and, for all it holds: . Thus, is an (additive) hypernear-ring. Let and define the hyperoperation as follows:
It follows that is a proper subhypergroup of and for a fixed it holds , for all . This implies that the mapping , defined by for any , is an inclusive homomorphism.
Now we will construct a left general additive hypernear-ring associated with an arbitrary hypergroup.
Theorem 6. Let be a hypergroup and . On the set define the hyperoperation and the operation as follows:
The obtained structure is a (left) general additive hypernear-ring.
Let . We prove that there exists such that for all . Let . Since we can choose and define . Obviously, . Now we prove that the hyperoperation is associative. Let Set and . Thus, if , then Thereby, for any , there exists such that . Define . Then, and for all it holds . Therefore, . So, . Similarly, we obtain that . Now, let . We prove that the equation has a solution . Since is a hypergroup, it follows that, for any , there exists such that . Define by . Then and . Similarly, we obtain that the equation has a solution in . We may conclude that is a hypergroup.
Obviously, is a semigroup, because the composition of functions is associative. Now we prove that the hyperoperation is left inclusively distributive with respect to the operation . Let . Set and . Let . Then, for all , it holds . Thus, , meaning that . □
For an arbitrary group G
, Malone and Heatherly [2
] denote by
the subset of
consisting of the functions which commute multiplicatively with the zero-function, i.e.,
is a sub-near-ring of
. The next result extends this property to the case of hyperstructures.
Let be a hypergroup with the identity element 0 (i.e., for all , it holds ), such that . Let . Then, is a subhypernear-ring of the general additive hypernear-ring .
Let . If , then , i.e., . Thus, . Let . We prove now that the equation has a solution . If we set and , where , for and , then and . Similarly the equation has a solution . Thus, is a subhypergroup of . Obviously, if , then it follows that , i.e., . So, is a subsemihypergroup of , implying that is a subsemihypernear-ring of . □
Let be an additive hypernear-ring such that is a proper subhypergroup of the hypergroup , having an identity element 0 satisfying the following properties:
, for all .
Then the hypernear-ring is –embeddable in the additive hypernear-ring .
For a fixed
, define a map
Obviously, . So, and, similarly as in the proof of Theorem 5, we obtain that the map defined by is an injective inclusion homomorphism. □
Example 4. On the set define an additive hyperoperation and a multiplicative operation having the Cayley tables described in Table 3 and Table 4, respectively: The structure is an (additive) hypernear-ring . Let . Then is a hypernear-ring (in particular it is a subhypernear-ring of ). Obviously, is a proper subhypergroup of the hypergroup , which has the identity 0 such that and , for all . It follows that, for each , is a map such that , for all ,
Clearly, the map , defined by , is an injective inclusion homomorphism, so the hypernear-ring R is W-embeddable in .
If is a group, then, for any , it holds and , meaning that the transformation near-ring of a group G is in fact the structure . Furthermore, if is a zero-symmetric near-ring, i.e., a near-ring in which any element x satisfies the relation , then the map ρ constructed in the proof of Theorem 8 is the injective homomorphism . Thus, according with Theorem 8, it follows that the zero-symmetric near-ring is W-embeddable in the near-ring , where is any group containing as a proper subgroup.
If is a group, then the following inclusions hold: , where both and are sub-(hyper)near-rings of the hypernear-ring . Considering now a hypergroup, the same inclusions exist: , but generally and are not subhypernear-rings of .
Let be a hypergroup with the identity element 0 (i.e., for all it holds ) such that . Let . Then, is a subhypernear-ring of the general hypernear-ring
Let . If , then it holds . Since , it follows that . Thus, . Let . We prove that the equation has a solution . If we set and , for all , then and, for all , it holds and , meaning that . Similarly, the equation has a solution in . So, is a subhypergroup of . Obviously, if , then . So, . Thus is a subsemihypergroup of . Therefore, is a subhypernear-ring of . □