Sorasak Leeratanavalee

The Order of Generalized Hypersubstitutions of Type

The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of type were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of type were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of type . In this paper, we use similar methods to characterize idempotent generalized hypersubstitutions of type and determine the order of each generalized hypersubstitution of this type. The main result is that the order is 1,2 or infinite.

Submonoids of generalized hypersubstitutions

In this paper we define the operation ©G on the set of all generalized hypersubstitutions and investigate some algebraic-structural properties of the set of all generalized hypersubstitutions and of some submonoids M of the set of all generalized hypersubstitutions, respectively.

All Regular Elements in Hyp G (2)

In this paper we consider mappings which map the binary operation symbol f to the term (f) which do not necessarily preserve the arities. We call these mappings generalized hypersubstitutions. Any generalized hypersubstitution can be extended to a mapping on the set of all terms of type = (2). We de ne a binary operation on the set (2) of all generalized hypersubstitutions of type = (2) by using this extension The set (2) together with the identity generalized hypersubstitution which maps f to the term f() forms a monoid. We determine all regular elements of this monoid.

Natural partial ordering on e(hypg(2))

The concept of generalized hypersubstitutions was introduced by S. Leeratanavalee and K. Denecke as a way of making precise the concepts of strong hyperidentity and M-strong hyperidentity. The set HypG(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. All idempotent and regular elements in the monoid of all generalized hypersubstitutions of type τ = (2) were studied by W. Puninagool and S. Leeratanavalee. In this paper, we determine all primitive idempotent elements of this monoid and characterize the natural partial ordering on the set of all idempotent of this monoid.

GENERALIZED DERIVED ALGEBRAS AND GENERALIZED INDUCED ALGEBRAS

Substituting for the fundamental operations of an algebra term operations we get a new algebra of the same type, called a generalized derived algebra. Such substitutions are called generalized hypersubstitutions. Generalized hypersubstitutions can also be applied to every equation of a fully invariant equational theory. The equational theory generated by the resulting set of the equations induces on every algebra of the type under consideration a fully invariant congruence relation. If we factorize the generalized derived algebra by this fully invariant congruence relation we will obtain an algebra which we call generalized induced algebra. In this paper, we prove some properties which transfer the starting algebras to generalized derived algebras and to generalized induced algebras.

Locally strong endomorphisms of paths

We determine the number of locally strong endomorphisms of directed and undirected paths-direction here is in the sense of a bipartite graph from one partition set to the other. This is done by the investigation of congruence classes, leading to the concept of a complete folding, which is used to characterize locally strong endomorphisms of paths. A congruence belongs to a locally strong endomorphism if and only if the number l of congruence classes divides the length of the original path and the points of the path are folded completely into the l classes, starting from 0 to l and then back to 0, then again back to l and so on. It turns out that for paths locally strong endomorphisms form a monoid if and only if the length of the path is prime or equal to 4 in the undirected case and in the directed case also if the length is 8. Finally some algebraic properties of these monoids are described.

All maximal completely regular submonoids of Hyp_{G}(2)

Abstract In this paper we consider mappings σ which map the binary operation symbol f to the term σ (f) which do not necessarily preserve the arity. These mapping are called generalized hypersubstitutions of type τ = (2) and we denote the set of all these generalized hypersubstitutions of type τ = (2) by HypG (2). The set HypG(2) together with a binary operation defined on this set and the identity generalized hypersubstitution which maps f to the term f(x1, x2) forms a monoid. In this paper, we determine all maximal completely regular submonoids of this monoid.

All maximal idempotent submonoids of HypG(2)

Abstract The purpose of this paper is to determine all maximal idempotent submonoids and some maximal compatible idempotent submonoids of the monoid of all generalized hypersubstitutions of type τ = (2).

The Order of Normal Form Generalized Hypersubstitutions of Type τ = (2)

In 2000, K. Denecke and K. Mahdavi showed that there are many idempotent elements in HypNφ(V ) the set of normal form hypersubstitutions of type = (2) which are not idempotent elements in Hyp(2) the set of all hypersubstitutions of type = (2). They considered in which varieties, idempotent elements of Hyp(2) are idempotent elements of HypNφ(V ). In this paper, we study the similar problems on the set of all generalized hypersubstitutions of type = (2) and the set of all normal form generalized hypersubstitutions of type = (2) and determine the order of normal form generalized hypersubstitutions of type = (2).

The Relationship between Some Regular Subsemigroups of

The concept of regular subsemigroups plays an important role in the theory of semigroup. In this work, we study the relationship between some regular subsemigroups on the monoid of all generalized hypersubstitutions of type .