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

Abstract

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.

State Machines and Hypergroups

“…However, it is possible for both associativity and inverted associativity to be valid in a magma. Such cases can be found in the examples of [63], which presents a detailed study of the left/right almost-hypergroups. For the quasi-canonical LA-hypergroups, see [64].…”

“…While there exists only one single element magma which is a group and also a LA/RA-group, there exist 3 4 = 81 magmas with 2 elements. These magmas can be constructed, classified and enumerated, with the techniques and methods which are developed in [69][70][71] and [63]. In the following propositions, these magmas are presented via their Cayley tables.…”

Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

“…Theorem 1 (see [19]). A ∩ B is LA-submodule of M, where A and B are the LA-submodules of an LA-module M. Definition 3 (see [9]).…”

“…rough their hyperideals and hypersystems, they investigate various important characterizations of LAhyperrings. Massouros and Yaqoob [19] presented the study of algebraic structures, left/right almost groups, and hypergroups equipped with the inverted associativity axiom, and they analyzed the algebraic properties of these special nonassociative hyperstructures. We refer readers to see if they want to learn more about LA-rings [9,[20][21][22][23].…”

On Cyclic LA-Hypergroups

“…The concept of LAhypergroups was given in [19] by Rehman and Yaqoob, where they explored some interesting properties of LA-hyperrings. In [20], Massouros and Yaqoob investigated some very useful and fundamental characteristics of LA-hypergroups.…”