Zero Semigroup Research Articles

The Vertex Distance on Cayley Digraphs of Rectangular Groups with respect to the Cartesian Product Connection Sets

A rectangular group S is a semigroup isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. The Cayley digraph Cay(S,A) of a finite rectangular group S with respect to a nonempty subset A of S is defined as a digraph with vertex set S and arc set consisting of ordered pairs (u,ua) in SxS for some a in A. The set A is called a connection set of Cay(S,A). Let u and v be in S. The distance from u to v in Cay(S,A), denoted by d(u,v), is the number of arcs in the shortest directed u-v path if one exists and is infinity otherwise. In this paper, we characterize the distances of any two vertices in Cayley digraphs of rectangular groups with respect to the Cartesian product connection sets

Hausdorff series in semigroup rings of rectangular bands

"The Hausdorff series provides a solution to the equation $w=\log(e^ue^v)$ given by a recursive formula which can be expressed as nested commutators of $u$ and $v$. Evolutions of the Haussdorff series in various algebras and rings has been considered in obtaining a closed form of this formula. We consider the rectangular band $L_m\times R_n$ determined by the left zero semigroup $L_m$ and the right zero semigroup $R_n$ of order $m$ and $n$, respectively. Let $\mathbb R\langle L_m\times R_n\rangle$ be the semigroup ring spanned on $L_m\times R_n$ together with the identity element $1$. We provide a closed form of the formula for solving the equation in $\mathbb R\langle L_m\times R_n\rangle$."

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.

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].

On the Independence Number of Cayley Digraphs of Rectangular Groups

A rectangular group is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. Some special cases of rectangular groups consisting of left groups and right groups are also considered here. Let $$ \mathrm {Cay}(S,A) $$ denote the Cayley digraph of the rectangular group S with the connection set A. In this paper, we are interested in studying some properties of $$ \mathrm {Cay}(S,A) $$ such as the independence, weakly independence, path independence, and weakly path independence. Furthermore, the independence numbers for those properties of $$ \mathrm {Cay}(S,A) $$ are also determined.

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.

Domination in Cayley Digraphs of Right and Left Groups

Let Cay(S,A) denote a Cayley digraph of a semigroup S with a connection set A. A semigroup S is said to be a right group if it is isomorphic to the direct product of a group and a right zero semigroup and S is called a left group if it is isomorphic to the direct product of a group and a left zero semigroup. In this paper, we attempt to nd the value or bounds for the domination number of Cayley digraphs of right groups and left groups. Some examples which give sharpness of those bounds are also shown. Moreover, we consider the total domination number and give the necessary and sufficient conditions for the existence of total dominating sets in Cayley digraphs of right groups and left groups.

ИНЪЕКТИВНЫЕ И ПРОЕКТИВНЫЕ ПОЛИГОНЫ НАД ВПОЛНЕ 0-ПРОСТОЙ ПОЛУГРУППОЙ

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

On left quasi-ample semigroups with ΠL1-embeddable band of idempotents

ABSTRACTLet ΠL1 denote a direct power of L1, the two-element left zero semigroup with identity adjoined. A semigroup S is called left quasi-ample if for each a∈S there exists a unique idempotent a+∈S such that for all x, y∈S1 and the left ample condition holds. Generalizing a recent result in [3], we prove that the semigroups in the title are embeddable into certain transformation semigroups. Our embedding provides an easy way to construct (finite) proper covers for (finite) such semigroups. Moreover, we show that each proper such semigroup is embeddable into a semidirect product of a ΠL1-embeddable band by a right cancellative monoid, giving a partial answer to a question raised in [1].

Infinite compact sets of idempotents in $$\beta S$$ β S

We show that if S is a countably infinite right cancellative semigroup and T is an infinite compact set of idempotents in the Stone–Cech compactification \(\beta S\) of S, then T contains an infinite compact left zero semigroup.

Fixed point compactifications

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that \(ap=pa=p\) for all \(p\in S\setminus G\) and \(a\in G\). Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted \(\theta G\). We show that if G is \(\sigma \)-compact, then \(\theta G\setminus G\) contains a copy of \(\beta \omega \setminus \omega \). In contrast, if G is not \(\sigma \)-compact, then \(\theta G\) is the one-point compactification.

Isomorphism Conditions for Cayley Graphs of Rectangular Groups

A rectangular band is defined as a direct product of a left zero semigroup and a right zero semigroup, and a rectangular group is defined as a direct product of a group and a rectangular band. In this paper, we give some equivalent conditions for Cayley graphs of a rectangular group to be isomorphic to each other.

On $$\mathcal {R}$$ R -unipotent semigroups with $$\Pi L^1$$ Π L 1 -embeddable band of idempotents

Let $$\Pi L^1$$ denote a direct power of $$L^1$$ , the two-element left zero semigroup with identity adjoined. We prove that the semigroups in the title are embeddable into transformation semigroups which naturally generalize the Vagner one-point completion of the symmetric inverse semigroup. We give some examples and show that our representation provides a convenient way to construct E-unitary covers and embeddings of E-unitary such semigroups into semidirect products of $$\Pi L^1$$ -embeddable bands by groups.

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.

Cycloidal algebras

Abstract 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.

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ϱ a congruence on S. Consider the sequence ϱ (0)⊆ϱ (1)⊆⋯⊆ϱ (n)⊆⋯ of congruences on S, where ϱ (0)=ϱ and, for an arbitrary non-negative integer n, ϱ (n+1) is defined by (a,b)∈ϱ (n+1) if and only if (xa,xb)∈ϱ (n) for all x∈S. We show that \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )\) for an arbitrary congruence ϱ on a semigroup S, where lrc(ϱ) denotes the least left reductive congruence on S containing ϱ. We focuse our attention on congruences ϱ on semigroups S for which the congruence \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is left reductive. We prove that, for a congruence ϱ on a semigroup S, \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is a left reductive congruence of S if and only if \(\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}\) is a left reductive congruence on the factor semigroup S/ϱ (here ι (S/ϱ) denotes the identity relation on S/ϱ). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S n =S n+1 is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a 1 , . . . , a n ) = c for some c ∈ A and all the a 1 , . . . , a n ∈ A) or a projection (i.e., f(a 1 , . . . , a n ) = a i for some i and all the a 1 , . . . , a n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x a = y a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a 2 for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.

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.

THE SEMIGROUPS OF BINARY SYSTEMS AND SOME PERSPECTIVES

Given binary operations and on a set X, define a product binary operation as follows: x2y := (x ∗ y) ◦ (y ∗ x). This in turn yields a binary operation on Bin(X), the set of groupoids defined on X turning it into a semigroup (Bin(X), 2)with identity (x∗y = x) the left zero semigroup and an analog of negative one in the right zero semigroup (x ∗ y = y). The composition 2 is a generalization of the composition of functions, modelled here as leftoids (x∗y = f (x)), permitting one to study the dynamics of binary systems as well as a variety of other perspectives also of interest. § 0. In what follows we associate with a set X a semigroup (Bin(X), 2)which generalizes in a rather natural way the composition of functions f : X → X. Classes of groupoids defined on a set X may form semigroups or even ideals and the axioms defining these classes may be analyzed according to the properties of these classes in (Bin(X), 2)for the set X. The fact that some sets of axioms produce subsemigroups of (Bin(X), 2)for the set X, whereas other sets do not, appears to be an interesting and possibly important distinction for the class under discussion. Some classes which appear to be new are also pointed out, including the linear groupoids over rings with identity, usually taken to be commutative. The strong bounded d-algebras are shown to be an interesting class described as a convenient intersection of three other classes each determined by one of the axioms. Given the standard equivalence these algebras will (indirectly) satisfy several M V -axioms. Other properties such as separability are looked at as well.

The subdirectly irreducible algebras in the variety generated by graph algebras

We show that every non-trivial subdirectly irreducible algebra in the variety generated by graph algebras is either a two-element left zero semigroup or a graph algebra itself. We characterize all the subdirectly irreducible algebras in this variety. From this we derive an example of a groupoid (graph algebra) that generates a variety with NP-complete membership problem. This is an improvement over the result of Z. Szekely who constructed an algebra with similar properties in the signature of two binary operations.