Hypernear-rings with a defect of distributivity

Abstract

Since in a near-ring the distributivity holds just on one side (left or right), it seems naturally to study the behaviour and properties of the set of elements that "correct" the lack of distributivity, in other words that elements that assure the validity of the distributivity. The normal subgroup of the additive structure of a near-ring generated by these elements is called a defect of distributivity of the near-ring. The purpose of this note is to initiate the study of the hypernear-rings (generalizations of near-rings, having the additive part a quasicanonical hypergroup) with a defect of distributivity, making a comparison with similar properties known for near-rings.

Division hypernear-rings

“…The first part of this section gathers together the main properties of division near-rings, most of them presented in Ligh's papers [18,19]. The theory of hypernear-rings with a defect of distributivity is briefly recalled in the second part of this section, using the authors' paper [14].…”

“…Keeping the terminology from our previous paper [14], by a hypernear-ring we mean an algebraic system (R, +, •), where R is a non-empty set endowed with a hyperoperation "+" : R×R −→ P * (R), and an operation "•" : R×R −→ R, satisfying the following axioms:…”

“…In the same period, the theory of hyperrings enriched itself with different types of hyperrings, having only one or both between addition and multiplication as hyperoperation (for example: additive hyperring [29], multiplicative hyperring [27], superring [22], hyperring in the general case [4,29]), or substituting the distributivity with the weak distributivity [29] (when the equality is replaced by non void intersection between the left and right side), or with the inclusive distributivity [12,16] (when the equality symbol is substituted by the inclusion one). A detailed discussion of this terminology is included in [13,14,24]. Even if the term "inclusive distributivity" is the most appropriate one for defining this property, it was used only in few papers, while " the weak distributivity" was preferred for the same type of distributivity.…”

“…In this paper, as in our previous one [14], by a hypernear-ring (R, +, •) we mean a quasicanonical hypergroup (R, +) endowed also with a multiplication, such that (R, •) is a semigroup with a bilaterally absorbing element, and the multiplication is inclusive distributive with respect to the hyperaddition on the left side. If the distributivity holds, i.e.…”

“…The authors have recently started [14] the study of hypernear-rings R with a defect of distributivity D, where D is the normal subhypergroup of the additive structure (R, +) generated by the elements…”