Corrections and Extensions in Left and Right Almost Semigroups

In this paper we elaborated the concept that on what conditions left almost semigroup (LA-Semigroup), right almost semigroup (RA-Semigroup) and a groupoid become commutative and further extended these results on medial, LA-Group and RA-Group. We proved that the relation of LA-Semigroup with left double displacement semigroup (LDD-semigroup), RA-Semigroup with left double displacement semigroup (RDD-semigroup) is only commutative property. We highlighted the errors in the recently developed results on LA-Semigroup and semigroup [17, 1, 18] and proved that example discussed in [18] is semigroup with left identity but not paramedial. We extended results on locally associative LA-Semigroup explained in [20, 21] towards LA-Semigroup and RA-Semigroup with left zero and right zero respectively. We also discussed results on n-dimensional LA-Semigroup, n-dimensional RASemigroup, non commutative finite medials with three or more than three left or right identities and finite as well as infinite commutative idempotent medials not studied in literature.

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].

Rings in which every left zero-divisor is also a right zero-divisor and conversely

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

Automorphism groups of semigroups of upfamilies

A family [Formula: see text] of nonempty subsets of a set [Formula: see text] is called an upfamily if for each set [Formula: see text] any set [Formula: see text] belongs to [Formula: see text]. The extension [Formula: see text] of [Formula: see text] consists of all upfamilies on [Formula: see text]. Any associative binary operation [Formula: see text] can be extended to an associative binary operation [Formula: see text]. In the paper, we study automorphisms of extensions of groups, finite monogenic semigroups, null semigroups, right zero semigroups and left zero semigroups. Also, we describe the automorphism groups of extensions of some semigroups of small cardinalities.

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.

The zero-divisor graph of a ring with involution

For a *-ring [Formula: see text], we associate a simple undirected graph [Formula: see text] having all nonzero left zero-divisors of [Formula: see text] as vertices and, two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. In case of Artinian *-rings and Rickart *-rings, characterizations are obtained for those *-rings having [Formula: see text] a complete graph or a star graph, and sufficient conditions are obtained for [Formula: see text] to be connected and also for [Formula: see text] to be disconnected. For a Rickart *-ring [Formula: see text], we characterize the girth of [Formula: see text] and prove a sort of Beck’s conjecture.

Answers to Some Questions Concerning Rings with Property (A)

A ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

Backtracking with cut via a distributive law and left-zero monoids

We employ the framework of algebraic effects to augment the list monad with the pruning cut operator known from Prolog. We give two descriptions of the resulting monad: as the monad of free left-zero monoids, and as a composition via a distributive law of the list monad and the ‘unary idempotent operation’ monad. The scope delimiter of cut arises as a handler.