The Order of Hypersubstitutions of Type (2, 2)

Thawhat Changphas,Klaus Denecke

doi:10.1007/s10114-005-0778-z

Abstract

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type (2, 2), i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type (2, 2). The main result is that the order is 1, 2, 3, 4 or infinite.

Highlights

X {x1, x2, x3, . . .} be a countably infinite alphabet of variables such that the sequence of the operation symbols fi i∈I is disjoint with X, and let Xn {x1, x2, . . . , xn} be an n-element alphabet where n ∈ Æ
The set of all terms of type τ over the alphabet X is defined as the disjoint union
Any mapping σ : {fi : i ∈ I} → Wτ X is called a ÝÔ Ö×Ù ×Ø ØÙØ ÓÒ of type τ if σ fi is an ni-ary term of type τ for every i ∈ I

Summary

Introduction

Any hypersubstitution of type [2, 1] has order either infinite or less than or equal to 3. Var b {x1} and Lemma 2.4 ii , we have σa,b has infinite order.

Results

Conclusion