Normal subobjects and semi-direct products in normally Mal'cev varieties
by
Francis Borceux
Universite de Louvain
Coauthors: Dominique Bourn

An algebraic theory which contains a unique constant 0 and a group operation yields a corresponding semi-abelian variety. Giving the group operation is equivalent to giving a Mal'cev operation p(x, y, z) which is associative, that is

p æ è x, y, p(u, v, z) ö ø =p æ è p(x, y, u), v, z ö ø .

This associativity condition is itself equivalent to the conjunction of both properties

p æ è x, t, p(t, y, z) ö ø =p(x, y, z),   p æ è p(x, y, t), t, z ö ø =p(x, y, z).

An algebraic theory which contains a unique constant 0 and a Mal'cev operation which satisfies only one of these last two axioms will be called ``normally Mal'cev''; the corresponding variety is still semi-abelian.

The first important property of ``normally Mal'cev varieties'' is the ``congruence uniformity'' property. Namely, if R is a congruence on an algebra A, the various equivalence classes are in bijection.

The second major property is the characterization of normal subobjects: in a semi-abelian category, a subobject B subset or equal A is normal when it is the kernel of some morphism. In the case of groups, this reduces to

a in A and b in B ===> a+b-a in B.

In the case of rings, this becomes

a in A and b in B ===> ab in B and ba in B.

These examples handle operations

\alpha1(x, y)=x+y-x,   \alpha2(x, y)=xy,   \alpha3(x, y)=yx

which all satisfy the axiom \alpha_i(x, 0)=0. Such an operation is called ``normal in y''. More generally, an operation

\alpha(x1, ... , xn, y1, ... , ym)

is called ``normal in y₁, ... , y_m'' when

\alpha(x1, ... , xn, 0, ... , 0)=0.

In a normally Mal'cev variety, a subobject B subset or equal A is normal when for every such operation \alpha normal in y₁, ... , y_m

a1, ... , an in A,  b1, ... , bm in B ===> \alpha(a1, ... , an, b1, ... , bm) in B.

This property somehow explains the terminology ``normally Mal'cev''.

Putting together the congruence uniformity property and the characterization of normal subobjects allows an elegant description of the semi-direct product in a normally Mal'cev variety. Let G be a fixed algebra. An action of G on an algebra B consists in giving (with straightforward axioms) an element

\alpha * (g1, ... , gn) * (b1, ... , bm) in B

for every operation \alpha(x₁, ... , x_n, y₁, ... , y_m) normal in y₁, ... , y_m and all elements g₁, ... , g_n in G, b₁, ... , b_m in B. The corresponding semi-direct product admits as underlying set the cartesian product B×G. Its structure of algebra is obtained in the following way. Given a n-ary operation \alpha of the theory, consider the operation

\alpha   (x1, ... , xn, y1, ... , yn) = \alpha(y1+x1, ... , yn+xn)-\alpha(x1, ... , xn).

where

x+y=p(x, 0, y) and x-y=p(x, y, 0),

with p(x, y, z) a normally Mal'cev operation of the theory. This operation [`(\alpha)] is normal in y₁, ... , y_n and the algebra structure of the semi-direct product is given by

\alpha æ è (b1, g1), ... , (bn, gn) ö ø = æ è \alpha   * (g1, ... , gn) * (b1, ... , bn), \alpha(g1, ... , gn) ö ø .

In the case of groups, it suffices to define the composition \alpha(x, y)=xy. The formula above, entirely symmetric in all variables, reduces then to the classical unsymmetric formula

(b, g)·(b', g') = æ è \alpha   * (g, g') * (b, b'), \alpha(g, g') ö ø . = æ è b+(g*b'), gg' ö ø .

Date received: May 20, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-19.