Quasivarieties and Varieties of Ordered Algebras: Regularity and Exactness

(Link - pdf).

Authors: Alexander Kurz (University of Leicester) and Jiří Velebil (Czech Technical University in Prague).

Reference: Retrieved from Alexander Kurz's homepage, 24 July 2013.

Comments: After last week's paper, which could only have been recommended to me based on my most recent publication, this week's paper seems to hark back to the very first paper I published back in 2005 when I was doing (more or less) pure mathematics for my Masters degree. That paper looked at coalgebra, which I have previously discussed in a post about a paper which shares an author with this one. This week's paper looks at the categorically dual (and perhaps more familiar) notion of algebra.

This paper fits within (or perhaps just to one side of) the mathematical field of universal algebra, which very broadly is the study of mathematics specified by equations. The power of equations is a theme I've come back to a few times on this blog, e.g. here. A class of mathematical structures, called algebras, that are specified by a certain set of equations is called a variety. For example, the class of all groups is a variety. A quasivariety is similar except that it is specified by conditional equations where each equation is conditioned by a list of equations. For example if we wanted to require that a symbol f be interpreted as an injective function then we do this by writing

(fx = fy) implies x=y.

The really interesting results in universal algebra involve describing (quasi)varieties in ways that are apparently unconnected to their definition by (quasi)equational logic. The one I'm most familiar with is Birkhoff's HSP theorem, which characterises (quasi)varieties by certain 'closure' conditions; this paper concerns an alternative characterisation where varieties are interpreted as categories, and we then enquire what structure such a category should possess.

So much for the standard picture; this paper picks up the suggestion of the 1976 paper Varieties of ordered algebras by Bloom which replaces equations, =, by inequations, ≤. That paper gave a Birkhoff-style characterisation of such 'ordered' (quasi)varieties; this paper gives these classes their categorical characterisation. For this to happen we need to relax our idea of what a category is; normal categories have a set of arrows between any two objects, called a hom-set, but the categories of this paper have a partial order on their hom-sets. This is a particular manifestation of a general idea called enriched category theory, where hom-sets can be replaced by objects from (almost) any category.

NOTE: Because of leave my next post will be in a fortnight.