We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products. transformation semigroup, permutation group, wreath product, Krohn-Rhodes Theory

Let iX, T) be a transformation group with compact Hausdorff phase space X. The points x and y of A" are said to be proximal provided, whenever ß is a member of the unique compatible uniformity of X, there exists i£ T such that (x/, yt) GßIf x and y are not proximal, they are said to be distal. Let P denote the proximal relation in X. P is a reflexive, symmetric, T invariant relation, but is not...

A ?semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. Schein 9] and Tamura 11] have investigated commutative ?semigroups, Trotter 13] exponential, Nagy 4] weakly exponential and Bonzini and Cherubini Spoletini 2] nite ?semigroups. The pupose of the present paper is to investigate archimedean weakly commutative and H?commutative ?semigroups. 1 Weakly comm...

Denote by T (X) the semigroup of full transformations on a set X. For ε ∈ T (X), the centralizer of ε is a subsemigroup of T (X) defined by C(ε) = {α ∈ T (X) : αε = εα}. It is well known that C(idX) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(idX) contains all non-invertible trans...

For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappingsOX on X modulo the full transformation semigroup TX . In other words, we ask what is the smallest cardinality of a set A of mappings such that 〈OX ∪A 〉 = TX . When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set...

We study the number of solutions of the general semigroup equation in one variable, X α = X β , as well as of the system of equations X 2 = X, Y 2 = Y, XY = Y X in H ≀ T n , the wreath product of an arbitrary finite group H with the full transformation semigroup T n on n letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concern...