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The most natural seminearrings, i.e., the seminearrings of all self-maps of additive semigroups are necessarily multiplicatively regular but they need not be additively regular. The purpose of this paper is to investigate additively regular seminearrings. We mainly focus on the study of congruences in various types of additively regular seminearrings such as additively inverse, additively Clifford and Bandelt seminearrings. We deduce that for a restricted type of additively inverse seminearrings there exists an inclusion preserving bijective correspondence between the set of all normal congruences and that of all full k-ideals. Finally, we characterize those seminearrings which are the subdirect product of a distributive lattice and a zero symmetric near-ring. [ABSTRACT FROM AUTHOR]

It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them all via filters. To achieve this, we introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). We study congruences which can be described via filters. We show that the lattice of semilattice (left normal band, right normal band) congruences is a homomorphic image of the lattice of negative (right negative, left negative) operation-compatible quasiorders. [ABSTRACT FROM AUTHOR]

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω- E-reflexive semigroups, are studied. (( ω)) and ((( ω))) are found to be the least Eω-Clifford and Eω- E-reflexive congruences on S, respectively. Characterizations of the congruences (( ω)) and ((( ω))) are developed. The lattices of all Eω-Clifford and Eω- E-reflexive congruences are also studied. [ABSTRACT FROM AUTHOR]

We consider a congruence ρ on a semigroup S as a subsemigroup of the direct product S × S. We prove that if ρ has finite derivation type (FDT), then so does S. [ABSTRACT FROM AUTHOR]

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C P(S) and the greatest elements of these sublattices are given. [ABSTRACT FROM AUTHOR]

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups. [ABSTRACT FROM AUTHOR]

We give an example of a tight inverse semigroup which is not bisimple and not congruence-free. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the congruence extension property and the ideal extension property for compact semigroups. We present a characterization of compact semigroups with the ideal extension property and prove that each compact semigroup with the congruence extension property also has the ideal extension property. [ABSTRACT FROM AUTHOR]

We investigate E[sup *]-dense semi\-groups as analogues of E-dense semigroups for semigroups with zero. We give a characterisation theorem for E[sup *]-dense semigroups whose idempotents form a *-rectangular band. The construction methods of generalised Rees matrix semigroups are employed to provide examples and illustrations. Our results are analogous to those of Weipoltshammer for E-dense semigroups. [ABSTRACT FROM AUTHOR]

A well known elementary argument shows that a totally disconnected normal subgroup of a connected topological group is central, in [21 K. H. HOFMANN has shown that by a skillful application of this argument (or variations of it) a much wider class of interesting centrality results can be obtained. In the present paper we offer a modification of the argument also applying to congruences on subsemigroups of topological groups. [ABSTRACT FROM AUTHOR]

On any regular semigroup S, the least group congruence σ, the greatest idempotent pure congruence τ and the least band congruence β are used to give the M-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (A,K,T), where K and T are the restrictions of the K-and T-relations on C(S) to A. Such triples are characterized abstractly and form the objects of a category M whose morphisms are surjective T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category M whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from M to M. Several properties of the classification of regular semigroups induced by this functor are established. [ABSTRACT FROM AUTHOR]

Presents a theory of existence varieties including appropriate notions of biidentities and biinvariant congruences. Biidentities for e-solid semigroups; Varities of orthogroups and orthogroup monoids; Bifree objects in varities of completely simple semigroups.