Clones of Finite Idempotent Algebras with Strictly Simple Subalgebras

Graduate Thesis Or Dissertation

Citations

Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/s7526c44f

Abstract

Abstract: We determine the clone of a finite idempotent algebra A that is not simple and has a unique nontrivial subalgebra S with more than two elements. Under these conditions, the proper subalgebras and the quotient algebra of A are finite idempotent strictly simple algebras of size at least 3 and it is known that such algebras are either affine, quasiprimal, or of a third classification. We focus on the first two cases. By excluding binary edge blockers from the relational clone when S is affine and by excluding ternary edge blockers from the relational clone together with an additional condition on the subuniverses of A² when S is quasiprimal, we give a nice description of the generating set of the relational clone of A. Thus, by the Galois connection between operations and relations, we determine the clone of A.

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