Inverse Congruence Research Articles

Structure theorems for weakly B-abundant semigroups

The aim of this article is to provide structure theorems for weakly B-abundant semigroups satisfying the Congruence Condition (C), where B is a band. Such semigroups may be thought of as generalisations of orthodox semigroups. Our focus is on providing a description of a weakly B-abundant semigroup S with (C) as a spined product of a weakly B-abundant semigroup SB (depending only on B) and S/γB, where γB is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form for γB. In addition, we build on an existing result of Ren to show that a weakly B-superabundant semigroup (C) is a semilattice of rectangular bands of monoids.

Semigroups which Have a minimum primitive inverse congruence

We investigate primitive inverse semigroups and Brandt semigroups as analogues of groups for semigroups with zero. We give a description of the minimum primitive inverse congruence on a categorical E *-dense E-semigroup. We show that a categorical semigroup S with a primitive inverse congruence has a minimum such congruence if $$\tilde D\left( S \right)$$ is *-dense in S. Here $$\tilde D\left( S \right)$$ is the least full, weakly self-conjugate, *-unitary and *-reflexive subsemigroup of S. Our results are analogous to those of Fountain, Pin and Weil for general semigroups.

On E-solid Semigroups with Inverse Transversals

We characterize E-solid semigroups with inverse transversals and some special such semigroups in terms of congruences. Some properties of inverse congruences over completely simple semigroups are obtained.

Primitive inverse congruences on categorical semigroups

AbstractWe give an abstract description of the kernel of a proper primitive inverse congruence on a categorical semigroup. More specifically, we show that it is a *-reflexive, *-unitary, *-dense subsemigroup, and that on a given categorical semigroup there is a one–one correspondence between such subsemigroups and the proper primitive inverse congruences. Our results allow us to give a description of the minimum proper primitive inverse semigroup congruence on a strongly E*-dense categorical semigroup.

Congruence extensions in regular semigroups

Let τ be a congruence on a full regular subsemigroup R of a regular semigroup S. The least congruence on S that contains τ is described. The description is in closed form modulo τ when R is self conjugate and τ extends to a congruence on S. Important congruences on S such as the least inverse and least orthodox congruences can be explicitly described by this approach in terms of extendable band congruences on the idempotent generated subsemigroup. Conditions for a congruence on R to extend to S are given. For the special case where S is E-solid, the least inverse congruence on S has an especially simple description.

On left quasinormal orthodox semigroups

SynopsisThe existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

On regular semigroups whose idempotents form a subsemigroup

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.