Dualisability versus Residual Character : a Theorem and a Counterexample

Content Provider	Semantic Scholar
Author	Davey, Brian A. Pitkethly, Jane G.
Copyright Year	2006
Abstract	We show that a finite algebra must be inherently non-dualisable if the variety that it generates is both residually large and congruence meetsemidistributive. We also give the first example of a finite dualisable algebra that generates a variety that is residually large. There is no obvious connection between the dualisability of a finite algebra and the residual character of the variety it generates. Certainly, there are many nondualisable algebras that generate a residually small variety: every finite algebra that does not have a near-unanimity term but generates a congruence-distributive variety [4, 12]. Nevertheless, there are many large classes of algebras for which it turns out that every finite member that generates a residually large variety is non-dualisable. As examples, there are the classes of groups [19, 8], commutative rings with identity [3, 8], bands [11, 9, 15], flat graph algebras [14, 13], p-semilattices [7] and closure semilattices [6, 13]. The weight of these examples led the first two authors to the following rash conjecture: ‘Every finite algebra that generates a residually large variety is non-dualisable’ [18]. This paper partially vindicates that conjecture. We show that a finite algebra must be inherently non-dualisable if the variety that it generates is both residually large and congruence meet-semidistributive (Corollary 3.3). In particular, the conjecture is true for every finite algebra with a semilattice reduct (Corollary 3.2). This paper also provides the first counterexample to the conjecture. In Section 4, we present a finite algebra that is dualisable and yet generates a variety that is residually large. Our counterexample is a term-reduct of a four-element ring, and the variety it generates is congruence permutable. 1. A semilattice-based example In this section, we study one particular three-element algebra, and show the relationship between a proof that it generates a residually large variety and a proof that it is inherently non-dualisable. This example provides some insight into the impetus for the main theorem, which is proved in Section 3. Roughly speaking, a finite algebra A is inherently non-dualisable if there is no natural representation for the quasivariety ISP(B), whenever B is a finite algebra such that A ∈ ISP(B). For a precise definition of inherent non-dualisability (indeed, 2000 Mathematics Subject Classification. Primary: 08A05; Secondary: 08B26, 06A12.
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Language	English
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Subject Keyword	Bands Class Closure Congruence of squares Exanthema Graph - visual representation Graph algebra Kleene algebra Mathematics Subject Classification Neoplasm Metastasis Ring device Speaking (activity) Zeller's congruence non-T, non-B adult acute lymphoblastic leukemia
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