We introduce the notions of a generalised category and of an inductive generalised category over a band. Our purpose is to describe a class of semigroups which we name weakly B-orthodox. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids. Here B denotes a band of idempotents; we note that if B is a semilattice then a weakly B-or...

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4, 6 + 4k, 87− 4k) and S(9, 3 + 9k, 85− 9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r 1 , r1r2 + r 2 1k, r3 − r 2 1k), k ∈ Z, r1, r2, r3 ∈ Z , r1 ≥ 2 and gcd(r1, r2) = gcd(r1, r3) = 1, and calculate their univers...

The Brauer semigroup Bn of partitions of a 2n-element set into two-element subsets, can be generalized in various ways. For example we may consider arbitrary partitions to obtain the semigroup Cn. The visualization of elements of these semigroups as chips leads us to the Temperley-Lieb semigroup TLn and its analogue TLCn. We study combinatorial properties of these four semigroups. For example, ...

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960’s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green’s relation J is tri...

The notion of a &Gamma-semigroup was introduced by Sen [8] in 1981. We can see that any semigroup can be considered as a &Gamma-semigroup. The aim of this article is to study the concept of (0-)minimal and max- imal ordered bi-ideals in ordered &Gamma-semigroups, and give some charac- terizations of (0-)minimal and maximal ordered bi-ideals in ordered &Gamma- semigroups analogous to the charact...

In this paper we analyze the so called word-problem for (finite) combinatorial 0-simple semigroups and matrix semigroups from the viewpoint of computational complexity.

One of the most natural approaches to the study of regular semigroups is to impose restrictions on the partial ordering of their idempotents (erg/^e/—jfe==e). The principal object of this note is to describe the structure of the classes of regular semigroups whose idempotents form a tree or a unitary tree, respectively (see Definition 1). We determine, among other things, a complete set of inva...

The number of finite semigroups increases rapidly with the number of elements. Since existing counting formulae do not give the complete number of semigroups of given order up to equivalence, the remainder can only be found by careful search. We describe the use of mathematical results combined with distributed Constraint Satisfaction to show that the number of non-equivalent semigroups of orde...

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Mathematics (2) 88 (1968), 128–160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, Algebraic theory of machines, I: Prime decomposition theorem for finite semigroups and machines, Transactions of the Ameri...

Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we prove conditions ensuring that a ...