| Abstract:
Thanks to the validity of the de Morgan laws, several varieties of
ortholattices enjoy the principle of duality. This implies, in particular,
that whenever an identity or an implication is true in all OML's, its dual
is also true in all OMLs. Naturally, one is tempted to ask whether one
can define the variety of all OMLs by an irredundant self-dual
set of
equations. In this talk, we make use the algebraic structure inherent in
OML's to give a conceptual proof that any finitely based variety of
orthomodular lattices has a self-dual description with only two identities.
Among other things, this includes the well-known example of the variety of
all Boolean algebras.
References:
-
D. Kelly and R. Padmanabhan, Self-dual bases for varieties of
orthomodular lattices, Algebra Universalis 50 (2003)
141–147.
-
G. F. McNulty, Minimum bases for equational theories of groups and
rings, Annals of Pure and Applied Logic 127 (2004) 131–153.
-
R. Padmanabhan and R. W. Quackenbush, Equational theory of algebras
with distributive congruences, Proc. Amer. Math. Soc. 41 (1973),
373–377.
-
A. Tarski, An interpolation theorem for irredundant bases of closure
structures, Discrete Math. 12 (1975), 185–192.
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