The Algebraic Structure of a Semigroup of Sets of Transformations with Restricted Range (original) (raw)

Four-part semigroups - semigroups of Boolean operations

Discussiones Mathematicae - General Algebra and Applications, 2012

Four-part semigroups form a new class of semigroups which became important when sets of Boolean operations which are closed under the binary superposition operation f + g := f (g, . . . , g), were studied. In this paper we describe the lattice of all subsemigroups of an arbitrary four-part semigroup, determine regular and idempotent elements, regular and idempotent subsemigroups, homomorphic images, Green's relations, and prove a representation theorem for four-part semigroups.

Associative Algebraic Structures in the Set of Boolean Functions and Some Applications in Automata Theory

IEEE Transactions on Computers, 1971

A class of linear algebraic structures over the set of Boolean functions is presented. Afterwards, from these structures a new one is studied-the right-weak algebra. By using the properties of this algebra, some new results are established about the commutativity of the superposition, the ideals, the annulators, and the properties of the iteration. There are also discussed some special problems like the "square rooting" of a switching function, some numerical evaluations (e.g., the number of functions which commute with f), and the practical significance of the algebraic results in switching theory and system engineering. In the last part, after the introduction of a new class of finite automata, some of their useful properties are established by using the properties of the elements of a right-weak algebra. Index Terms-Annulators, coincidence, commutative Boolean algebra, ideals, right-weak linear algebra, square root of a Boolean function.

On a Semigroup of Sets of Transformations with Restricted Range

Thai Journal of Mathematics, 2016

This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restricted range Y X. We introduce the semi- group TP (X; Y ) of all non-empty subsets of T(X; Y ) under the operation AB := fab : a 2 A;B 2 Bg. We determine the idempotent and regular ele- ments in TP (X; Y ) for the case that jY j = 2. In particular, we characterize the (maximal) regular subsemigroups of TP (X; Y ), the largest semiband, and the (maximal) idempotent subsemigroups of TP (X; Y ).

On the Semigroup of Difunctional Binary Relations

FUDMA JOURNAL OF SCIENCES

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

© Hindawi Publishing Corp. GENERALIZED JOIN-HEMIMORPHISMS ON BOOLEAN ALGEBRAS

2001

We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boolean equivalence, and open filters. 2000 Mathematics Subject Classification: 06E25, 03G99. 1. Introduction. In [4]

On some semigroups generated from Cayley functions

Any transformation on a set S is called a Cayley function on S if there exists a semigroup operation on S such that β is an inner-translation. In this paper we describe a method to generate a semigroup with k number of idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.

Some Properties of Semigroups Generated from a Cayley Function

2019

Any transformation on a set S is called a Cayley function on S if there exists a semigroup operation on S such that β is an inner-translation. In this paper we describe a method to generate a semigroup with k number of idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.

Analysis of some Quasigroup transformations as Boolean Functions

MASSEE International Congress on …, 2009

Two kind of attacks on cryptographic primitives -the linear and the differential cryptanalysis, have been occupied the attention of the cryptographic community since several years ago. Every cryptographic primitive can be examined as a vector valued Boolean function. The prop ratio tables and the correlation matrices are important tools for analyzing the resistance of any Boolean function to the linear and the differential cryptanalysis. There are several cryptographic primitives based on the so called quasigroup transformations, and in this paper we analyze these quasigroup transformations as Boolean functions. We examine correlation matrices, prop ratio tables and some other cryptographic properties.