Implicit algebraic operations

Implicit algebraic operations

[Todd Wilson]

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 <= b <= c in a
distributive lattice, there is at most one b' such that

     b /\ b' = a   and   b \/ b' = 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' = 1    and    m' * m = 1.

It follows that inverses, when they exist, are also preserved by monoid
homomorphisms.

Now, the investigation of these partial operations gets one quickly into
non-surjective epimorphisms, dominions in the sense of Isbell, algebraic
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 problems?


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

Re: Implicit algebraic operations

[mhebert]

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
>
>