Journal of Pure and Applied Algebra 216 (2012) 2737–2753
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Journal of Pure and Applied Algebra
journal homepage: www.elsevier.com/locate/jpaa
Effective dimension of finite semigroups
, Benjamin Steinberg
Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden
Department of Mathematics; City College of New York, Convent Avenue at 138th Street, New York, NY 10031, United States
a r t i c l e i n f o
Received 23 July 2011
Received in revised form 7 April 2012
Available online 23 May 2012
Communicated by M. Sapir
MSC: 16G99; 20M10
a b s t r a c t
In this paper we discuss various aspects of the problem of determining the minimal
dimension of an injective linear representation of a finite semigroup over a field. We outline
some general techniques and results, and apply them to numerous examples.
© 2012 Elsevier B.V. All rights reserved.
Most representation theoretic questions about a finite semigroup S over a field k are really questions about the
semigroup algebra kS. One question that is, however, strictly about S itself is the minimum dimension of an effective linear
representation of S over k, where by effective we mean injective; we call this the effective dimension of S over k. Note that
semigroups (and in fact groups) with isomorphic semigroup algebras can have different effective dimensions. For example,
the effective dimension of Z/4Z over C is 1, whereas the effective dimension of Z/2Z × Z/2Z is 2 (since C has a unique
element of multiplicative order two), although both groups have algebras isomorphic to C
There are two natural questions that arise when considering the effective dimension of finite semigroups:
(a) Is the effective dimension of a finite semigroup decidable?
(b) Can one compute the effective dimension of one’s favorite finite semigroups?
These are two fundamentally different questions. The first question asks for a Turing machine that on input the Cayley table
of a finite semigroup, outputs the effective dimension over k. The second one asks for an actual number. Usually for the
second question one has in mind a family of finite semigroups given by some parameters, e.g., full (partial) transformation
monoids, full linear monoids over finite fields, full monoids of binary relations, etc. One wants to know the effective
dimension as a function of the parameters.
The effective dimension of groups (sometimes called the minimal faithful degree) is a classical topic, dating back to the
origins of representation theory. There doesn’t seem to be that much work in the literature on semigroups except for the
paper  of Kim and Roush and previous work  of the authors. This could be due in part to the fact that the question is
much trickier for semigroups because semigroup algebras are rarely semisimple. Also, minimal dimension effective modules
need not be submodules of the regular representation.
Question (a) has a positive answer if the first order theory of the field k is decidable. Indeed, to determine the effective
dimension of a finite semigroup one just needs to solve a finite collection of systems of equations and inequations over k
because the effective dimension is obviously bounded by the size of the semigroup plus one. Classical results of Tarski imply
E-mail addresses: email@example.com (V. Mazorchuk), firstname.lastname@example.org (B. Steinberg).
mazor/ (V. Mazorchuk), http://www.sci.ccny.cuny.edu/
benjamin/ (B. Steinberg).
0022-4049/$ – see front matter © 2012 Elsevier B.V. All rights reserved.