@MISC{Algebras_undecidabilityof, author = {Residuated Boolean Algebras}, title = {Undecidability of the Equational Theory of Some Classes of}, year = {} }

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Abstract

We show the undecidability of the equational theories of some classes of BAOs with a non-associative, residuated binary extra-Boolean operator. These results solve problems in Jipsen [9], P ra t t [21] and Roorda [22], [23]. This paper complements Andreka-Kurucz-Nemeti-Sain-Simon [3] where the emphasis is on BAOs with an associative binary operator. In this paper we are concerned with the decision problem for the equational theories of certain varieties of Boolean Algebras with residuated Operators (residuated BAOs for short). We recall that an n-ary (n> 0) operation / on a Boolean Algebra (BA from now on) A is an operator if it is additive (distributes over joins) in each of its arguments. A unary operator / on A is residuated if there is an operation g (called the conjugate of / ) such that for all x, y G A (where A is the universe of A) f(x) • y = 0 iff x- g{y) = 0. If / is n-ary, then it is called residuated if it is residuated in each of its arguments. We note that additivity follows from being residuated, and that two operations being conjugates of each other is an equational property (in fact expressible by positive equations). For more on (residuated) BAOs see Jipsen [9], [10], Jonsson [12], Jonsson-Tsinakis [14] and Jonsson-Tarski [13]. We use [9] as our main reference but we define all notions we are going to use. The classes of algebras in the present paper will be subclasses of the variety UR of unital residuated Boolean groupoids. These are residuated BAOs with a normal binary operator o together with its right and left conjugates> and <, respectively, and a constant E, the neutral element for o. (An operator is normal if its value is 0 whenever one of its arguments is 0.) In more detail, an algebra A = (A0,o,>,<,£) is in UR if Ao is a BA, o,>,< are binary operations and E is a constant satisfying a;o0 = 0 o x = 0,