A characterization of Commutative Semigroups

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup is u – inverse semigroup. We will also prove that if (S,.) is a H - semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

A right zero semigroup is left-cancellative

We prove the proposition addressed in the title of this paper.

Hausdorff series in a semigroup ring

Let [Formula: see text] and [Formula: see text] be the semigroup rings spanned on the right zero semigroup [Formula: see text], and on the left zero semigroup [Formula: see text], respectively, together with the identity element [Formula: see text]. We suggest a closed formula solving the equation [Formula: see text] which is the evolution of the Campbell–Baker–Hausdorff formula given by the Hausdorff series [Formula: see text] where [Formula: see text], in the algebras [Formula: see text] and [Formula: see text].

Independent domination number in Cayley digraphs of rectangular groups

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

HUBUNGAN DERIVASI PRIME NEAR-RING DENGAN SIFAT KOMUTATIF RING

Near-rings are generalize from rings. A research on near-ring is continous included a research on prime near-rings and one of this research is about derivation on prime near-rings. This article will reviewing about relation between derivation on prime near-rings and commutativity in rings with literature review method. The result is prime near-rings are commutative rings if a nonzero derivation d on N hold one of this following conditions: (i) , (ii) , (iii) , (iv) , (v) , (vi) , for all , with is non zero semigroup ideal from .

Cycloidal algebras

In this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *)n = (X; ∘), where x ∘ y = [x * y]n = x * [x * y]n−1, [x * y]0 = y. For several classes of examples we study the cycloidal index (m, n) of (X; *), where (X; *)m = (X; *)n for m > n and m is minimal with this property. We show that (X; *) satisfies the left cancellation law, then if (X; *)m = (X; *)n, then also (X; *)m−n = (X; *)0, the right zero semigroup. Finite algebras are shown to have cycloidal indices (as expected). B-algebras are considered in greater detail. For commutative rings R with identity, x * y = ax + by + c, a, b, c ∈ ℝ defines a linear product and for such linear products the commutativity condition [x * y]n = [y * x]n is observed to be related to the golden section, the classical one obtained for ℝ, the real numbers, n = 2 and a = 1 as the coefficient b.

CAYLEY AUTOMATON SEMIGROUPS

In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite.

On 0-homology of categorical at zero semigroups

The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg-MacLane homology of semigroups are given.

Local Homomorphisms of Topological Groups

A mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

PRESENTATIONS FOR SEMIGROUPS AND SEMIGROUPOIDS

We consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [17], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite subsemigroupoids. We apply this result to extend the results of [6], giving characterizations of finite generation and finite presentability in Rees matrix semigroups over semigroupoids.