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In the past decade, category theory has widened its scope and now
inter acts with many areas of mathematics. This book develops some
of the interactions between universal algebra and category theory
as well as some of the resulting applications. We begin with an
exposition of equationally defineable classes from the point of
view of "algebraic theories," but without the use of category
theory. This serves to motivate the general treatment of algebraic
theories in a category, which is the central concern of the book.
(No category theory is presumed; rather, an independent treatment
is provided by the second chap ter.) Applications abound throughout
the text and exercises and in the final chapter in which we pursue
problems originating in topological dynamics and in automata
theory. This book is a natural outgrowth of the ideas of a small
group of mathe maticians, many of whom were in residence at the
Forschungsinstitut fur Mathematik of the Eidgenossische Technische
Hochschule in Zurich, Switzerland during the academic year 1966-67.
It was in this stimulating atmosphere that the author wrote his
doctoral dissertation. The "Zurich School," then, was Michael Barr,
Jon Beck, John Gray, Bill Lawvere, Fred Linton, and Myles Tierney
(who were there) and (at least) Harry Appelgate, Sammy Eilenberg,
John Isbell, and Saunders Mac Lane (whose spiritual presence was
tangible.) I am grateful to the National Science Foundation who
provided support, under grants GJ 35759 and OCR 72-03733 A01, while
I wrote this book."
This paper is one of a series in which the ideas of category theory
are applied to problems of system theory. As with the three
principal earlier papers, [1-3], the emphasis is on study of the
realization problem, or the problem of associating with an
input-output description of a system an internal description with
something analogous to a state-space. In this paper, several sorts
of machines will be discussed, which arrange themselves in the
following hierarchy: Input process Machine Output process (Tree
automaton) Machine ~ ~ State-behavior Machine I Adjoint Machine
.(Sequential Machine) ., I Decomposable Machine (Linear System,
Group Machine) Each member of the hierarchy includes members below
it; examples are included in parentheaes, and each example is at
its lowest possible point in the hierarchy. There are contrived
examples of output process machines and IV state-behavior machines
which are not adjoint machines [3], but as yet, no examples with
the accepted stature of linear systems [4], group machines [5, 6],
sequential machines [7, Ch. 2], and tree automata [7, Ch. 4].
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