State-Morphism Algebras - General Approach

TL;DR

This paper provides a comprehensive theoretical framework for state-morphism algebras, including classifications of subdirectly irreducible cases and embeddings into diagonal algebras, with applications to various algebraic structures.

Contribution

It introduces a general theory of state-morphism algebras, characterizes subdirectly irreducible cases, and describes generators for related algebraic varieties.

Findings

Classification of subdirectly irreducible state-morphism algebras

Embedding of these algebras into diagonal algebras

Generators for varieties of state-morphism algebras

Abstract

We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is an idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of varieties of state-morphism algebras, in particular ones of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.