View Full Document

MINIMAL CLONES WITH ABELIAN REPRESENTATIONS



View the full content.
View Full Document
View Full Document

1 views

Unformatted text preview:

MINIMAL CLONES WITH ABELIAN REPRESENTATIONS KEITH A KEARNES Abstract We show that a minimal clone has a nontrivial abelian representation if and only if it is isomorphic to a minimal subclone of a finite cyclic group As an application we show that a minimal clone contains a Mal cev operation if and only if it is isomorphic to the clone of idempotent operations of a group of prime order 1 Introduction A clone is trivial if every operation is projection onto a variable A clone is minimal if it is not trivial but its only subclone is trivial Any minimal clone is generated by a single operation If C is a clone generated by the operation f then an f representation of C is a pair A hA f A i where A is a set f A is an operation on A and the assignment f 7 f A extends to a clone homomorphism from C to the concrete clone of operations on A generated by f A Such a representation will be called trivial if the clone of hA f A i is a trivial clone i e if f A is projection onto a variable A representation is faithful if the assignment f 7 f A extends to a clone isomorphism from C to the clone of operations on A generated by f A The size of the representation hA f A i is A Let C be a minimal clone and let f be an operation generating C The class of f representations of C is a variety of algebras which we denote by V The clone of V is C which is a minimal clone by assumption and the clone of each A V is either trivial or it is a minimal clone If we change from f to a different operation generating C then we get a term equivalent variety In this paper we shall often drop the reference to f when we speak of representations and consequently we shall only consider algebras and varieties up to term equivalence The term condition for an algebra A cf Chapter 3 of 3 is the assertion that for all operations t x y in Clo A and all a b A u v An the following implication holds t a u t a v t b u t b v 1 2 KEITH A KEARNES An abelian algebra is one which satisfies the term condition When using



Access the best Study Guides, Lecture Notes and Practice Exams

Loading Unlocking...
Login

Join to view MINIMAL CLONES WITH ABELIAN REPRESENTATIONS and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view MINIMAL CLONES WITH ABELIAN REPRESENTATIONS and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?