A distributive multisemilattice of type n is an algebra with a family of n binary semilattice operations on a common carrier that are mutually distributive. This concept for n=2 comprises the distributive bisemilattices (or quasilattices), of which distributive lattices and semilattices with duplicated operations are the best known examples. Multisemilattices need not satisfy the absorption law, which holds in all lattices.Kalman has exhibited a subdirectly irreducible distributive bisemilattice which is neither a lattice nor a semilattice. It has three elements. In this paper it is shown that all the subdirectly irreducible distributive multisemilattices are derived from those for n=2 simply by duplicating their operations in all possible ways. Thus, up to isomorphism there are of type n, but up to the coarser relation of polynomial equivalence there are only three. Hence every distributive multisemilattice is the subdirect product of irreducibles, each with two or three elements.The rest of the paper is devoted to the varieties of distributive multisemilattices. The lattice of these varieties is described, and bases for their identities are given.
CONTENTS1. Introduction........................................................................................................................52. Definition, basic examples and properties of multisemilattices...........................................63. The subdirectly irreducibles..............................................................................................134. The lattice of subvarieties of .................................................................................185. Subvarieties of defined by identities involving at most two operation symbols......246. Some further comments and open problems....................................................................34References...........................................................................................................................40
1985 Mathematics Subject Classification: Primary 06A12, 08B15; Secondary 05C40, 08B05.
@book{bwmeta1.element.zamlynska-553ea7ba-6a08-4b20-8ee7-4c4ed1ce33b2,
author = {Arthur Knoebel and Anna Romanowska},
title = {Distributive multisemilattices},
series = {GDML\_Books},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
address = {Warszawa},
year = {1991},
zbl = {0719.06002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-553ea7ba-6a08-4b20-8ee7-4c4ed1ce33b2}
}
Arthur Knoebel; Anna Romanowska. Distributive multisemilattices. GDML_Books (1991), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-553ea7ba-6a08-4b20-8ee7-4c4ed1ce33b2/