Multi-Hypersubstitutions and Colored Solid Varieties

TL;DR

This paper extends hypersubstitution theory to multi-hypersubstitutions and colored solid varieties, providing new tools to analyze the complex lattice of algebraic varieties.

Contribution

It introduces multi-hypersubstitutions and colored solid varieties, linking them to the equational theory of Universal Algebra and offering new methods to study algebraic lattice structures.

Findings

Developed the theory of multi-hypersubstitutions.

Connected colored terms with algebraic varieties.

Provided a new approach to analyze complex algebraic lattices.

Abstract

Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.