scholarly journals ON THE IMBEDDING OF A FINITE COMMUTATIVE SEMIGROUP OF IDEMPOTENTS IN A UNIQUELY FACTORABLE SEMIGROUP

1956 ◽  
Vol 42 (10) ◽  
pp. 772-775
Author(s):  
M. W. Weaver
Keyword(s):  
Commutative Semigroup ◽  
Finite Commutative Semigroup

scholarly journals On a problem of Baayen and Kruyswijk

1968 ◽  
Vol 16 (2) ◽  
pp. 145-149
Author(s):  
D. A. Burgess
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Finite Semigroup ◽  
Multiplicative Semigroup ◽  
Residue Classes ◽  
Finite Commutative Semigroup

We shall call a finite semigroup S arithmetical if there exists a positive integer N and a monomorphism μ of S into the multiplicative semigroup RN of the ring of residue classes of the integers modulo N. In 1965 P. C. Baayen and D. Kruyswijk [1] posed the problem' Is every finite commutative semigroup arithmetical? ' The purpose of this paper is to answer this question.

Note on the Davenport constant of the multiplicative semigroup of the quotient ring 𝔽p[x] 〈f(x)〉

2016 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
Haoli Wang ◽  
Lizhen Zhang ◽  
Qinghong Wang ◽  
Yongke Qu
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Multiplicative Semigroup ◽  
Davenport Constant ◽  
Prime Field ◽  
Group Of Units ◽  
Polynomial Formula ◽  
Finite Commutative Semigroup

Let [Formula: see text] be a finite commutative semigroup. The Davenport constant of [Formula: see text], denoted [Formula: see text], is defined to be the least positive integer [Formula: see text] such that every sequence [Formula: see text] of elements in [Formula: see text] of length at least [Formula: see text] contains a proper subsequence [Formula: see text] with the sum of all terms from [Formula: see text] equaling the sum of all terms from [Formula: see text]. Let [Formula: see text] be a polynomial ring in one variable over the prime field [Formula: see text], and let [Formula: see text]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring [Formula: see text] and proved that, for any prime [Formula: see text] and any polynomial [Formula: see text] which factors into a product of pairwise non-associate irreducible polynomials, [Formula: see text] where [Formula: see text] denotes the multiplicative semigroup of the quotient ring [Formula: see text] and [Formula: see text] denotes the group of units of the semigroup [Formula: see text].

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.

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