Algebra Universalis Structural entailment

We give a number of characterizations of structural entailment. In particular, we show that an alter ego M∼ structurally entails an algebraic relation s on a finite algebra M if and only if s can be obtained via a local construct from M∼ . We show, via a range of applications, that, whereas entailment is important in the study of dualisability, structural entailment is important in the study of full and strong dualisability. We also give an application to the transfer of strong dualities that connects this paper to our earlier paper [9] on full versus strong duality. The concept of structural entailment has been around since the birth of the theory of natural dualities in 1980 (see Davey and Werner [11]). Nevertheless, until now its weaker cousin, entailment, has received all of the attention. It is significant that, until this paper and its companion [9], the concept did not even have a name. In their seminal paper [8] on the syntax and semantics of entailment, Davey, Haviar and Priestley characterised entailment but did not consider structural entailment at all. In this paper we redress the balance. Once we state and prove the Structural Entailment Theorem (in Section 2) and consider its many applications (in Sections 3 and 4) it becomes clear why structural entailment hardly rated a mention in earlier work on entailment. Previously, authors considering entailment have been motivated by questions concerned with dualisability. Our results make it clear that, while structural entailment is important in connection with dualisability, its most important applications arise in connection with strong dualisability and the more inscrutable full dualisability. In Section 1, we give a brief introduction to natural duality theory tailored to our particular needs. Those familiar with dualisability, full dualisability, strong dualisability and entailment may wish to begin with Section 2, where structural entailment is defined and characterised, referring to Section 1 as needed. 1. Dualisability and all that it entails Here we shall give a very brief refresher on the basics of natural dualities. The definitions of dualisability, full dualisability and strong dualisability will be recalled, Presented by R. W. Quackenbush. Received June 27, 2003; accepted in final form May 12, 2005. 2000 Mathematics Subject Classification: 08C05; 08C15, 18A40.