Lattices of Quasi-Equational Theories as Congruence Lattices of Semilattices with Operators: Part I

نویسندگان

  • Kira V. Adaricheva
  • James B. Nation
چکیده

We show that for every quasivariety K of relational structures there is a semilattice S with operators such that the lattice of quasiequational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+, 0, F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found. 1. Motivation and terminology Our objective is to provide, for the lattice of quasivarieties contained in a given quasivariety (Q-lattices in short), a description similar to the one that characterizes the lattice of subvarieties of a given variety as the dual of the lattice of fully invariant congruences on a countably generated free algebra. Just as the result for varieties is more naturally expressed in terms of the lattice of equational theories, rather than the dual lattice of varieties, so it will be more natural to consider lattices of quasi-equational theories rather than lattices of quasivarieties. The basic result is that the lattice of quasi-equational theories extending a given quasi-equational theory is isomorphic to the congruence lattice of a semilattice with operators preserving join and 0. These lattices support a natural quasi-interior operator, the properties of which lead to new restrictions on lattices of quasi-equational theories. This is the first paper in a series of four. Part II shows that if S is a semilattice with both 0 and 1, and G is a group of operators on S, then there is a quasi-equational theory T such that Con(S,+, 0,G) is isomorphic to the lattice of quasi-equational theories extending T. The third part [30] shows that if S is any semilattice with operators, then ConS is isomorphic to the lattice of implicational theories extending some given implicational theory, but in a language that may not include equality. The fourth paper, Date: October 28, 2009. 1991 Mathematics Subject Classification. 08C15, 08A30, 06A12.

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عنوان ژورنال:
  • IJAC

دوره 22  شماره 

صفحات  -

تاریخ انتشار 2012