scholarly journals Radikal Prima Bi-Ideal Dalam Semiring Ternari

2019 ◽  
Vol 9 (2) ◽  
pp. 78
Author(s):  
Ratna Sari Widiastuti
Keyword(s):  
Commutative Semigroup ◽  
Ternary Semiring

A ternary semiring is an additive commutative semigroup with a ternary multiplication which satisfying some condition. This paper will be discuss about prime bi-ideal radikal in semiring ternary with definition and some theorem. If B is a bi-ideal in semiring ternary T, then radical prime bi-ideal of B in semiring ternary T is an intersection of all prime bi-ideal in semiring ternary T which containing B.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

Semigroup actions on ordered groupoids

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Niovi Kehayopulu ◽  
Michael Tsingelis
Keyword(s):  
Commutative Semigroup ◽  
Equivalence Relation ◽  
Semigroup Actions ◽  
Ordered Groupoid ◽  
Quasi Order

AbstractIn this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S̃ acting on the ordered groupoid G̃:=(G × S)/ρ̄ in such a way that G is embedded in G̃. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S̄ acts on an ordered groupoid Ḡ in such a way that G is embedded in S̄, then the ordered groupoid G̃ can be also embedded in Ḡ. We denote by ρ̄ the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ −1.

scholarly journals Finite AG-groupoid with left identity and left zero

2001 ◽  
Vol 27 (6) ◽  
pp. 387-389 ◽  
Author(s):  
Qaiser Mushtaq ◽  
M. S. Kamran
Keyword(s):  
Commutative Semigroup ◽  
Algebraic Structure ◽  
Zero Element ◽  
Commutative Group ◽  
Left Identity ◽  
Left Zero ◽  
Left Invertive Law

A groupoidGwhose elements satisfy the left invertive law:(ab)c=(cb)ais known as Abel-Grassman's groupoid (AG-groupoid). It is a nonassociative algebraic structure midway between a groupoid and a commutative semigroup. In this note, we show that ifGis a finite AG-groupoid with a left zero then, under certain conditions,Gwithout the left zero element is a commutative group.

ON THE HYPERSTABILITY OF A PEXIDERISED -QUADRATIC FUNCTIONAL EQUATION ON SEMIGROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 459-470 ◽  
Author(s):  
IZ-IDDINE EL-FASSI ◽  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Quadratic Functional Equation ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Image Position

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$, where the map $\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of $G$ with $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all $x\in G$. We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the $\unicode[STIX]{x1D70E}$-quadratic and $\unicode[STIX]{x1D70E}$-Drygas equations.

Extension of a Semigroup Embedding Theorem to Semirings

1975 ◽  
Vol 18 (2) ◽  
pp. 297-298 ◽  
Author(s):  
Paul H. Karyellas
Keyword(s):  
Commutative Semigroup ◽  
Embedding Theorem ◽  
Additive Semigroup ◽  
Left And Right

It is well known [1,3] that a commutative semigroup (S, +) can be embedded in a semigroup which is a union of groups if and only if S is separative (2a = a + b = 2b implies a = b). We extend this result to additively commutative semirings.A semiring (S, +, ⋅) is a set S with associative addition (+) and multiplication (⋅), the latter distributing over addition from left and right. In what follows (S, +, ⋅) will denote a semiring in which the additive semigroup (S, +) is commutative. An element 0 can be adjoined, where s = s + 0, 0 = 0 ⋅ s = s ⋅ 0 for all s in S, to form S°.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

scholarly journals An introduction ofF-graphs, a graph-theoretic representation of natural numbers

1992 ◽  
Vol 15 (2) ◽  
pp. 313-317 ◽  
Author(s):  
E. J. Farrell
Keyword(s):  
Commutative Semigroup ◽  
Unit Element ◽  
Natural Numbers ◽  
Graph Theoretic

A special type of family graphs (F-graphs, for brevity) are introduced. These are cactus-type graphs which form infinite families under an attachment operation. Some of the characterizing properties ofF-graphs are discussed. Also, it is shown that, together with the attachment operation, these families form an infinite, commutative semigroup with unit element. Finally, it is shown thatF-graphs are graph-theoretical representations of natural numbers.

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