Middle-East Journal of Scientific Research 23 (6): 1059-1063, 2015 ISSN 1990-9233 © IDOSI Publications, 2015 DOI: 10.5829/idosi.mejsr.2015.23.06.22259 Corresponding Author: R. Elageili, Department of Mathematics, Benghazi University, Benghazi, Libya. 1059 Subalgebras of the Free Heyting Algebra on One Generator R. Elageili Department of Mathematics, Benghazi University, Benghazi, Libya Abstract: In this paper we describe the subalgebras of the free Heyting algebra A on one generator, generated 1 by an arbitrary single element of A and we give a general theorem which provides an explicit classification of 1 all the subalgebras of A . 1 Key words: Free Heyting algebra Subalgebra INTRODUCTION In contrast with Boolean algebras, finitely Heyting algebras are a generalization of Boolean wasshown by Mckinsey and Tarski in the 1940s algebras; the most typical example is the lattice of open ([1]). For one generator, A is well understood sets of a topological space. It is well known that Heyting (The Rieger Nishimura ladder), but from two on, the algebras are algebraic models of intuitionistic structure remains mysterious, though manyproperties are propositional logic, which is properly contained in known. classical propositional logic. Heyting algebras are special With the help of recursively described distributive lattices and they form a variety. Kripkemodels H , Bellissima ([2]) gave a Definition: An algebra A = (A, , , , 0, 1) is a Heyting due to Grigolia ([3]) and Esakia. algebra bounded distributive lattice with least element 0 The free Heyting algebra A on one generators may and greatest element 1 and For all x, y A, x y is the be defined as the Lindenbaum algebra ofintuitionistic greatest element z of A such that x z y, (wkere is propositional logic IPC on a set P = P of one defined by: x y if and only if x y = x). This element x propositional variable, this is theso-called `Rieger- y is called the pseudo-complement of x with respect to. Nishimura lattice', or `ladder', [4, 5], has an explicit In any Heyting algebra the pseudo-complement x description as a lattice; allelements which are not its least of x is defined by x = x 0. Note that x x = 0 and or greatest elements 0,1, lie in antichains of size 2, of that x is the greatest element having this property. which thereare countably many arranged in levels and A complete Heyting algebra is a Heyting algebra that the partial order relation between these is quite easilyand is a complete lattice (every subset has a supremum). explicitly described. Let V be a variety. Recall that an algebra A V is said Our aim in this paper is to provide an explicit to be a free algebra over V, if there exists a set E A such classification of all the subalgebras of A . that E generates A and every mapping from E to an algebra B V can be extended uniquely to a Definition: Let A be an algebra, X A. The subalgebra homomorphism from A to B. In this case E is said to be a generated by X is the intersection of allalgebras set of free generatorsof A. If E is finite then A is said to containing A written X . (This can be constructed by be a finitely generated free algebra. We denote a finitely closing up X under the operations.) generated free Heyting algebra with free generators The Rieger-Nishimura Ladder, A, is shown in (which is uniquelydetermined up to isomorphism) by A . Figure 1. Free Heyting algebras arise naturally as the Lindenbaumalgebras of intuitionistic propositional logic Subalgebras of A : First I give the subalgebras of A (IPC) with propositional variables over the emptytheory. which are generated by only one element. generated free Heyting algebras are infinite as 1 representationof A . Essentially the same construction is 1 1 1 1 1 1