A representation theory for the variety generated by the triangle

Abstract

The relational structure (Ws; N) and the algebra (W~; A, V) are both referred to as the triangle. The variety, W3, generated by (W3; A, V) was first studied in E. FRIED and G. GR~TZ~R [7]; it is the first member of an important chain of varieties of weakly associative lattices (see E. FRIED [6]). Gditzer and Fried introduced W 3 as a natural generalization of the class of distributive lattices and showed that many important universal algebraic properties of distributive lattices extend to W~. It is therefore reasonable to ask whether there is a topological representation theory for W3 similar to H. A. PRIESTLEY's representation theory for (bounded) distributive lattices [10, 11]. We shall see that the answer is 'Yes'. We add a nullary operation 9 to the operations on Ws, and then establish a duality between the variety A generated by A=(Wa; A, V, . ) and a category X of structured compact spaces. The algebras in A are pointed Ws-algebras, and the stiucture on the objects of X consists of a pseudo-order p and a nullary operation , . The only subdirectly irreducible (bounded) distributive lattice is the two-element chain, and consequently every (bounded) distributive lattice L can be embedded into 2 x for some set X, or equivalently, into the lattice N(X) of all subsets of X. Part of the role of PRmSTLEY'S duality, or STONE'S duality [13] before it, was to specify the set X and to identify those subsets of X which form a lattice isomorphic to X. A similar line of reasoning applies to the variety A and the algebra A. The three nontrivial subalgebras of A are the only subdirectly irreducibles in A and hence every A-algebra, B, can be embedded into A x for some set X. It is easily seen that A x is isomorphic to the set of 3-block partitions of X (allowing empty blocks) pseudo-ordered by