20 research outputs found
Boole's Method I. A Modern Version
A rigorous, modern version of Boole's algebra of logic is presented, based
partly on the 1890s treatment of Ernst Schroder
The lattice of varieties of implication semigroups
In 2012, the second author introduced and examined a new type of algebras as
a generalization of De Morgan algebras. These algebras are of type (2,0) with
one binary and one nullary operation satisfying two certain specific
identities. Such algebras are called implication zroupoids. They invesigated in
a number of articles by the second author and J.M.Cornejo. In these articles
several varieties of implication zroupoids satisfying the associative law
appeared. Implication zroupoids satisfying the associative law are called
implication semigroups. Here we completely describe the lattice of all
varieties of implication semigroups. It turns out that this lattice is
non-modular and consists of 16 elements.Comment: Compared with the previous version, we rewrite Section 3 and add
Appendixes A and
Studies in Logic and the Foundations of Mathematics
p. 341–349In this paper a characterization of principal congruences of De Morgan algebras is given and from it we derive that the variety of De Morgan algebras has DPC and CEP. The characterization is then applied to give a new proof of Kalman's characterization of subdirectly irreducibles in this variety and thus to obtain the representation theorem for DeMorgan algebras first proved by Kalman and independently, using topological methods, by Bialynicki-Birula and Rasiowa. From this representation it is deduced that finite De Morgan algebras are not the only ones with Boolean congruence lattices. Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice