Nikola Ruskuc

Partially Well-Ordered Closed Sets of Permutations

It is known that the “pattern containment” order on permutations is not a partial well-order. Nevertheless, many naturally defined subsets of permutations are partially well-ordered, in which case they have a strong finite basis property. Several classes are proved to be partially well-ordered under pattern containment. Conversely, a number of new antichains are exhibited that give some insight as to where the boundary between partially well-ordered and not partially well-ordered classes lies.

Regular closed sets of permutations

Machines whose main purpose is to permute and sort data are studied. The sets of permtutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets to be enumerated by rational generating functions. As a consequence we give the first non-trivial examples of pattern closed sets of permutations all of whose closed subclasses have rational generating functions.

Sorting with two ordered stacks in series

The permutations that can be sorted by two stacks in series are considered, subject to the condition that each stack remains ordered. A forbidden characterisation of such permutations is obtained and the number of permutations of each length is determined by a generating function.

GENERATORS AND FACTORISATIONS OF TRANSFORMATION SEMIGROUPS

If E is the set of idempotents and G the group of units within a full transformation semigroup ℐ x , then EG = GE = ℐ x if X is finite. The question of identifying the subsemigroup EG = GE = 〈G∪E〉 in the case where X is infinite leads to an investigation of interrelations among various naturally occurring subsemigroups of ℐ X . In the final section it is shown that precisely two additional elements µ, v are needed in order that G∪E∪{µ, v} should generate ℐ x .

AUTOMATIC SEMIGROUPS WITH SUBSEMIGROUPS OF FINITE REES INDEX

The notion of automaticity has been widely studied in groups and some progress has been made in understanding this notion in the wider context of semigroups. The purpose of this paper is to study t...

REWRITING A SEMIGROUP PRESENTATION

Let be a finitely presented semigroup having a minimal left ideal L and a minimal right ideal R. The main result gives a presentation for the group R∩L. It is obtained by rewriting the relations of , using the actions of on its minimal left and minimal right ideals. This allows the structure of the minimal two-sided ideal of to be described explicitly in terms of a Rees matrix semigroup. These results are applied to the Fibonacci semigroups, proving the conjecture that S(r, n, k) is infinite if g.c.d.(n, k)>1 and g.c.d.(n, r+k−1)>1. Two enumeration procedures, related to rewriting the presentation of into a presentation for R∩L, are described. The first enumerates the minimal left and minimal right ideals of , and gives the actions of on these ideals. The second enumerates the idempotents of the minimal two-sided ideal of .

Generating the full transformation semigroup using order preserving mappings

For a linearly ordered set

SUBSEMIGROUPS OF THE BICYCLIC MONOID

X

On defining groups efficiently without using inverses

we consider the relative rank of the semigroup of all order preserving mappings

Computing Transformation Semigroups

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