Free Ortholattices

Abstract

It has been known for some time but does not seem to be anywhere in the literature that the variety of all ortholattices is generated by its finite members (see (4.2) of this paper). This is well known to imply that the word problem for free ortholattices is solvable. On the other hand, it is also known that the solution obtained this way is of no practical use. The main purpose of this paper is to present a workable solution.

The variety generated by the truth value algebra of type-2 fuzzy sets

“…Since this variety is locally nite, it would follow that any algebra satisfying equations (1){ (7) would be locally nite. But ortholattices satisfy these equations, and not all ortholattices are locally nite [2]. A similar argument holds when we restrict to the operations t; u since the variety V((M; u; t)) must also be locally nite, and not all the algebras satisfying equations (1){ (4) are locally nite, as all lattices satisfy (1){ (4), but not all lattices are locally nite.…”

Algebraic Aspects of Orthomodular Lattices

“…The third is easily seen from the above construction of free ortholattices. The fourth is found in [8]. Another very useful fact, easily proved along the lines of (3.4) is the following.…”

“…Again, the reader is directed to [8] for a complete proof, but we sketch the details. Given a set X, take another set X' in bijective correspondence with X and disjoint from X.…”

Gentzen Methods in Quantum Logic

“…For algebraic and semantical decision procedures, the reader is referred to [Bruns, 1976], [Goldblatt, 1974] and [Goldblatt, 1975]. Fortunately, minimal quantum logic enjoys these three kinds of decision procedures.…”