Monotone clones and the varieties they determine
type: Item
instance of: scholarly article
title: Monotone clones and the varieties they determine
zbMATH Open document ID: 0734.08001
DOI: 10.1007/BF00383763
published in: Order
publication date: 1990-01-01T00:00:00Z
review text: A clone C on a set P is a family of finitary functions which contains the projections and is closed under composition. A clone C is called monotone if there is some order \(\leq\) on P such that C is the set of all finitary order-preserving functions with respect to this order. A finite nontrivial algebra P is said to be order-primal if the clone of finitary functions on P is monotone. It is proved that the variety generated by an order-primal algebra is congruence-distributive iff it contains at most two non-isomorphic subdirectly irreducible algebras (Theorem 1.7). Further, part (ii) of Theorem 3.3 states that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories then the corresponding ordered sets or their duals generate that same order variety. The authors also show that every finite bounded ordered set of length 3 has a 5-ary order-preserving near unanimity function (see Theorem 4.4).
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4214084
zbMATH Keywords: monotone clone
zbMATH Keywords: congruence-distributive variety
zbMATH Keywords: category equivalence
zbMATH Keywords: order-primal algebra
zbMATH Keywords: near unanimity function
author: Brian A. Davey
author: Robert W. Quackenbush
author: Dietmar Schweigert
reviewed by: V. N. Saliĭ
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214084