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