Projective Bichains
نویسندگان
چکیده
An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice satisfying Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.
منابع مشابه
Varieties of Birkhoff Systems Part I
A Birkhoff system is an algebra that has two binary operations · and +, with each being commutative, associative, and idempotent, and together satisfying x · (x + y) = x + (x · y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bottom part of the lattice of subvarieties o...
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