Another Consequence of AP in Residually Small, Congruence Modular Varieties
Year: 1988
Type: article
Abstract: . Let be a congruence modular variety satisfying (C2) whose two generated free algebra is finite. If has AP and RS then has (R). Here (C2) and (R) are two properties defined in terms of the commutator in modular varieties. In this paper we continue the investigation of residually small, congru-ence modular varieties with the amalgamation property. In [1] we showed that RS + AP • CEP in a congruence modular variety that satisfied 3 crucial properties: (C2), (R) and 4-finiteness (defined below). The pur-pose of this paper is to show that the assumption that the variety satisfies (R) follows from the remaining ones. Following the proof of Corollary 9, we present a detailed example that may aid in the comprehension of the arguments. We use fairly standard notation of universal algebra. Algebras are denoted by boldface latin letters and their underlying sets by the lightface equivalent. Con(A) denotes the lattice of congruences of the algebra A, with smallest and largest congruences 0 and 1. Congruences are represented by lower case Greek letters. c • A/3, c • V/3 and [c•,/3] are the meet, join and commutator of the congruences • and,J. In a congruence modular variety, the commutator is a binary operation with a number of powerful properties. See [2, Section 4] or [4, Section 6] for the details and proofs. We say that an algebra A has property (C2) if for all c•, • C Con(A), [c•,/3] = c • A/3 A [1,1], and has property (R) if for all subalgebras B of A, [1^, 1^] IB = [lB, lB]. A variety]2 has (C2) or (R) just in case every member has the property. We comment hat [1^, I^]IB> [lB, lB] holds for all algebras A and subalgebras B, thus (R) asserts the reverse inclusion. We say a variety]2 is n-finite if the]Y-free algebra on n generators is finite, where n is a natural number. If f: A- • B is a homomorphism
Author: Clifford Bergman
Institution: Iowa State University
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W306660646