On the lattice of biorder ideals of regular rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

U-Concordant Semigroups

We introduce the relations [Formula: see text] and [Formula: see text] with respect to a subset U of idempotents. Based on [Formula: see text] and [Formula: see text], we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong’s work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence.

Free idempotent generated semigroups over bands and biordered sets with trivial products

For any biordered set of idempotents [Formula: see text] there is an initial object [Formula: see text], the free idempotent generated semigroup over[Formula: see text], in the category of semigroups generated by a set of idempotents biorder-isomorphic to [Formula: see text]. Recent research on [Formula: see text] has focused on the behavior of the maximal subgroups. Inspired by an example of Brittenham, Margolis and Meakin, several proofs have been offered that any group occurs as a maximal subgroup of some [Formula: see text], the latest being that of Dolinka and Ruškuc, who show that [Formula: see text] can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any [Formula: see text] lie in subgroups. However, little else is known of the “global” properties of [Formula: see text], other than that it need not be regular, even where [Formula: see text] is a semilattice. The aim of this paper is to deepen our understanding of the overall structure of [Formula: see text] in the case where [Formula: see text] is a biordered set with trivial products (for example, the biordered set of a poset) or where [Formula: see text] is the biordered set of a band [Formula: see text]. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given rise to a deep and fruitful research area. The class of abundant semigroups extends that of regular semigroups in a natural way and itself is contained in the class of weakly abundant semigroups. Our main results show that (1) if [Formula: see text] is a biordered set with trivial products then [Formula: see text] is abundant and (if [Formula: see text] is finite) has solvable word problem, and (2) for any band [Formula: see text], the semigroup [Formula: see text] is weakly abundant and moreover satisfies a natural condition called the congruence condition. Further, [Formula: see text] is abundant for a normal band [Formula: see text] for which [Formula: see text] satisfies a given technical condition, and we give examples of such [Formula: see text]. On the other hand, we give an example of a normal band [Formula: see text] such that [Formula: see text] is not abundant.

Completely simple congruences on E-inversive semigroups

We study completely simple congruences on an arbitrary [Formula: see text]-inversive semigroup [Formula: see text]. In particular, we show that every such congruence [Formula: see text] on [Formula: see text] is uniquely determined by its kernel and trace, and that the trace of [Formula: see text] is a congruence on the biordered set [Formula: see text]. Moreover, we investigate the complete lattice of all completely simple congruences on [Formula: see text] and show that the trace relation is a complete congruence on this lattice. We also construct a family of completely simple congruences on [Formula: see text].

CONSTRUCTION OF A R-STRONGLY UNIT REGULAR MONOID FROM A REGULAR BIORDERED SET AND A GROUP

In this paper we give a detailed study of R-strongly unit regular monoids. The relations between the biordered set of idempotents and the group of units in unit regular semigroups are better identified here. Conversely, starting from a regular biordered set E and a group G we construct a R-strongly unit regular semigroup S for which the set of idempotents E(S) is isomorphic to E as a biordered set and the group of units G(S) is isomorphic to G. The conditions to be satisfied by G and E are also listed.

PERIODIC ELEMENTS OF THE FREE IDEMPOTENT GENERATED SEMIGROUP ON A BIORDERED SET

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.

Idempotent-separating extensions of regular semigroups

For a regular biordered setE, the notion ofE-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered setE, anE-diagram in a categoryCis a collection of objects, indexed by the elements ofEand morphisms ofCsatisfying certain compatibility conditions. With such anE-diagramAwe associate a regular semigroupRegE(A)havingEas its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application ofRegE(A)to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a groupE-diagramA. In this paper, the necessary and sufficient condition for the existence of an extension ofSbyAis provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.

Biordered sets are biordered subsets of idempotents of semigroups

A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.