Summary: ON MULTIVARIETIES AND MULTIALGEBRAIC CATEGORIES
JiŸ r' i Ad' amek, JiŸ r' i Rosick' y
Dedicated to the 60 th birthday of Bill Lawvere.
Abstract. Multivarieties are classes of algebras presented by exclusive­or's of equa­
tions. A full characterization of categories which are equivalent to multivarieties
is presented, close to Lawvere's characterization of varieties. A comparison with
Y.Diers' concept of multialgebraic category is presented: this is precisely a mul­
tivariety with effective equivalence relations. Besides, multialgebraic categories are
shown to be precisely those categories which can be sketched by a (finite product,
coproduct)­sketch.
I. Introduction
The aim of the present paper is to compare and characterize two related gener­
alizations of the concept of a (finitary) variety: multivariety, introduced in [AHR],
and multialgebraic category, studied by Y.Diers [D 1 ]. A multivariety has as syntax
multiequations, i.e., exclusive­or's of equations. An algebra A satisfies a multiequa­
tion
5
i2I
(ff i = fi i )
(where 5 stands for ``exclusive or'') provided that for every interpretation of vari­

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

Collections: Computer Technologies and Information Sciences