Identities in Multivalued Algebraic Structures

Massouros, Ch. G.; Massouros, G. G.

doi:10.1063/1.3498355

Cited by 5 publications

(11 citation statements)

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“…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.…”

Section: Identities and Symmetric Elementsmentioning

confidence: 99%

“…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.…”

Section: Identities and Symmetric Elementsmentioning

confidence: 99%

“…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…”

Section: Example 8 the Cayley Tablesmentioning

confidence: 99%

“…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].…”

Section: Example 8 the Cayley Tablesmentioning

confidence: 99%

“…See [32] for the origin of the terminology. 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}.…”

Section: Proposition 14mentioning

confidence: 99%

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

Massouros

Yaqoob

2021

Mathematics

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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.

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Section: Identities and Symmetric Elementsmentioning

confidence: 99%

Section: Identities and Symmetric Elementsmentioning

confidence: 99%

“…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…”

Section: Example 8 the Cayley Tablesmentioning

confidence: 99%

“…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].…”

Section: Example 8 the Cayley Tablesmentioning

confidence: 99%