For a non-empty set X denote the full transformation semigroup of by T(X). Let \(\sigma\) be an equivalence relation on and E(X, \(\sigma\)) denotes (under composition) all \(\alpha\) : \(\mapsto\) X, such that \(\subseteq\) ker(\(\alpha\) ). Semigroup transformations with restricted occur when we take whose kernel is contained in some fixed equivalence, \(\sigma\)). First, found disjoint union...

Mitsch’s natural partial order on the semigroup of binary relations has a complex relationship with the compatible partial order of inclusion. This relationship is explored by means of a sublattice of the lattice of preorders on the semigroup. The natural partial order is also characterised by equations in the theory of relation algebras. 1. Natural partial orders Having formulated a characteri...

Let Q be a finite set and let k be a non negative integer. A (partial) function f: Q + Q is a k-map if ]sf’] < k, Vq E Q. A transformation semigroupx = (Q, S) is a k-t.s. if each s ES is a k-map. Let TS be the collection of all (finite) transformation semigroups and let c: TS + N be the complexity function. For background on complexity see [2]. The main theorem of this paper shows that if X is ...

This paper provides a new generalization of fuzzy finite state machines, fuzzy transformation semigroups and their relationship. Consider a cubic structure, we introduce cubic finite state machines, cubic transformation semigroups, cubic successor, cubic exchange properties cubic subsystems, cubic submachines, cubic q-twins, cubic retrievable and study fundamental properties of them. We provide...

Stream ciphers are often used in applications where high speed and low delay are a requirement. The Solitaire stream cipher was developed by B. Schneier as a paper-and-pencil cipher. Solitaire gets its security from the inherent randomness in a shuffled deck of cards. In this paper we investigate semigroups and groups properties of the Solitaire stream cipher and its regular modiÞcations.

A (partial) transformation α on the finite set {1, . . . , n} moves an element i of its domain a distance of |i − iα| units. The work w(α) performed by α is the sum of all of these distances. We derive formulae for the total work w(S) = ∑ α∈S w(α) performed by various semigroups S of (partial) transformations. One of our main results is the proof of a conjecture of Tim Lavers which states that ...

in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

There are many ways to construct hierarchical decompositions of transformation semigroups. The holonomy algorithm is especially suitable for computational implementations and it is used in our software package. The structure of the holonomy decomposition is determined by the action of the semigroup on certain subsets of the state set. Here we focus on this structure, the skeleton, and investiga...

This work aims at understanding the structure of process algebras via holonomy decomposition. In that connection, the work studies the skeleton of the transformation semigroup which is obtained from the natural transition relation between the processes of process algebra, and observes that it is of height one.

We decompose tensor products of the defining representation of a Cartan type Lie algebra W (n) in the case where the number of tensoring does not exceed the rank of the Lie algebra. As a result, we get a kind of Schur duality between W (n) and a finite dimensional non-semisimple algebra, which is the semi-group ring of the transformation semigroup Tm .