Re: Help me remember this definition?
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Sun, 3 Jul 2005 21:03:27 +0000 (UTC)
In article <1120423300.845434.284690@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Snis Pilbor <snispilbor@xxxxxxxxx> wrote:
> I recall reading at one point, a definition for a very general
>sort of algebraic structure. One basically had the underlying set, and
>then one had an arbitrary number of k-ary relations defined on it (with
>possibly many different k's) and arbitrarily many "axioms" which were
>just equations using these relations.
This is a general or universal algebra.
> For example groups are a special case: a 2-ary multiplication, a
>1-ary inverse, and a 0-ary identity. Associativity is an equation in
>just the 2-ary,
It is technically called an "identity", not an equation.
> and identity/inverses together are an equation in the
>1-ary and the 0-ary (ie, x*inverse(x)=identity).
> Now suddenly I could really use a definition just like that... but
>I can't remember the exact definition, the exact name of the
>generalized structures, nor what book I read this in!
Any book whose title is "Universal algebra" will have what you
want. The two classical references are "Universal Algebra" by
P.M. Cohn and the book of the same title by Gratzer. There is also the
GTM book "A course in universal algebra" by Stanley Burris and
H.P. Sankappanavar; that one was long out of print, but is available
in a new edition at
http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
There is also "Algebras, Lattices, Varieties, vol. 1" (affectionately
known as "ALVIN", by Ralph McKenzie, George McNulty, and Walter
Taylor; as far as I know, however, vols. 2 or higher do not yet exist
in print.
As usual I will plug my advisor's book, George Bergman's "An
invitation to General Algebra and Universal Constructions", which you
can download at
http://math.berkeley.edu/~gbergman/245/
The definitions you are looking for are in Chapter 8.
The precise definitions may vary in detail. Here is the sequence of
definitions from Burris-Sankappanavar (pp. 25-26 in the millenium
edition):
DEF 1.1 For A a nonempty set and n a nonnegative integer we define
A^0={emptyset}, and for n>0, A^n is the set of n-tuples of elements
from A. An n-ary operation (or function) on A is any function f from
A^n to A; n is the arity (or rank) of f. A finitary operation is an
n-ary operation for some n. The image of <a_1,...,a_n> under an
n-ary operation f is denoted by f(a_1,...,a_n). An operation f on A
is called a nullary operation (or constant) if its arity is zero; it
is completely determined by the image f(emptyset) in A of the only
element emptyset in A^0, and as such it is convenient to identify it
with the element f(emptyset). Thus, a nullary operation is thought
of as an element of A. An operation f on A is unary, binary, or
ternary if its arity is 1, 2, or 3, respectively.
DEF 1.2. A language (or type) of algebras is a set F of function
symbols such that a nonnegative integer n is assigned to each member
f of F. This integer is called the arity (or rank) of f, and f is
said to be an n-ary funciton symbol. The subset of n-ary function
symbols in F is denoted by F_n.
DEF 1.3 If F is a language of algebras then an algebra A of type "F"
[script in the original] is an ordered pair <A,F> where A is a
nonempty set and F is a family of finitary operations on A indexed
by the language "F" such that corresponding to each n-ary function
symbol f in "F" there is an n-ary operation f^A on A. The set A is
called the universe (or underlying set) of <A,F>, and the f^A's are
called the fundamental operations of A.
Here are the definitions in Bergman's book:
DEF 8.1.1. A type will mean a pair Omega = (|Omega|,ari_Omega) where
|Omega| is a set, and ari_Omega is a map from |Omega| to sets. The
elements s in |Omega| are called the operation symbols of Omega, and
for each such s, the set ari(s) is called the arity of the operation
symbol s. Omega is called finitary if all of its operation symbols
have finite arity.
DEF. 8.1.2 If Omega is a type, an algebra of type Omega will mean a
pair A = (|A|, (s_A)_{s in |Omega|}), where |A| is a set, and for
each s in |Omega|, s_A is an ari(s)-ary operation on |A|.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
.
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