On Using the Join Operation to Define Classes of Algebras

Abstract

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.

Key Words:

• Join-irreducible
• Order-preserving
• Prevariety
• Quasivariety
• Semigroup

Mathematics Subject Classification:

• Primary 08C15
• Secondary 20M20, 20M07

Notes

Communicated by D. Easdown.