Re: Implicit algebraic operations

["mhebert" <mhebert@aucegypt.edu>]

Hi everyone,

Todd Wilson asks :
> My question is this: Does a definitive treatment of this phenomenon [pa=
rtial operations
, ...,  non-surjective epimorphisms,...] in
> "algebraic" categories exist? Are there still some mysteries/open probl=
ems?

It seems to me that the problem of
"characterizing the algebraic theories giving rise to varieties where all=
 the epis are surjective"
(posed by Bill Lawvere in
Some algebraic problems in the context..., LNM 61 (1968) )
is still essentially open. Anyone knows otherwise?
(A "classical" version might be to find a - syntactic-  condition on the =
equations necessary and sufficient to have all epis surjective in its cat=
egory of its models)

Michel Hebert




Fromcat-dist@mta.ca

Tocategories@mta.ca

Cc

DateSun, 26 Nov 2006 22:16:59 -0800

Subjectcategories: Implicit algebraic operations



> I was going through some of my old notes today and came across
> investigations I had done several years ago on implicit operations in
> Universal Algebra. These are definable partial operations on algebras
> that are preserved by all homomorphisms. Here are two examples:
>
> (1) Pseudocomplements in distributive lattices. Given a <=3D b <=3D c i=
n a
> distributive lattice, there is at most one b' such that
>
> b /\ b' =3D a and b \/ b' =3D c.
>
> Because lattice homomorphisms preserve these inequalities and equations=
,
> the uniqueness of pseudocomplements implies that, when they exist, they=

> are also preserved by homomorphisms.
>
> (2) Multiplicative inverses in monoids. Similarly, given an element m
> in a monoid (M, *, 1), there is at most one element m' such that
>
> m * m' =3D 1 and m' * m =3D 1.
>
> It follows that inverses, when they exist, are also preserved by monoid=

> homomorphisms.
>
> Now, the investigation of these partial operations gets one quickly int=
o
> non-surjective epimorphisms, dominions in the sense of Isbell, algebrai=
c
> elements in the sense of Bacsich, implicit partial operations in the
> sense of Hebert, and other topics. Some of the references that I know
> about are listed below.
>
> My question is this: Does a definitive treatment of this phenomenon in
> "algebraic" categories exist? Are there still some mysteries/open probl=
ems?
>
>
> REFERENCES
>
> PD Bacsich, "Defining algebraic elements", JSL 38:1 (Mar 1973), 93-101.=

>
> PD Bacsich, "An epi-reflector for universal theories", Canad. Math.
> Bull. 16:2 (1973), 167-171.
>
> PD Bacsich, "Model theory of epimorphisms", Canad. Math. Bull. 17:4
> (1974), 471-477.
>
> JR Isbell, "Epimorphisms and dominions", Proc. of the Conference on
> Categorical Algebra, La Jolla, Lange and Springer, Berlin 1966, 232-246=
.
>
> JM Howie and JR Isbell, "Epimorphisms and dominions, II", J. Algebra, 6=

> (1967), 7-21.
>
> JR Isbell, "Epimorphisms and Dominions, III", Amer. J Math. 90:4 (Oct
> 1968), 1025-1030.
>
> M Hebert, "Sur les operations partielles implicites et leur relation
> avec la surjectivite des epimorphismes", Can. J. Math. 45:3 (1993), 554=
-575.
>
> M Hebert, "On generation and implicit partial operations in locally
> presentable categories", Appl. Cat. Struct. 6:4 (Dec 1998), 473-488.
>
> --
> Todd Wilson A smile is not an individual
> Computer Science Department product; it is a co-product.
> California State University, Fresno -- Thich Nhat Hanh
>
>