SIGNATURE (UNIVERSAL ALGEBRA)

In mathematics, a 'signature' for an algebraic structure ''A'' over an underlying set ''S'' is a list of the operations that characterize ''A'', along with their arities. Signatures are a key concept in universal algebra, and are also employed in model, category, and type theory.
A signature consists of two lists, each usually enclosed by ⟨ and ⟩, whose items are separated by commas. One list begins with ''S'' , followed by the symbols for the operations characterizing ''A''. An operation ''f'' of arity ''n'', where ''n'' is a natural number, is a function ''f'': ''S''ⁿ→''S''. Distinguished members of ''S'', such as identity elements, are treated as operations of arity 0. The arities make up the second list, called the ''type'' of ''A''. The arities are listed in the same order as the corresponding operations. To list the operations defining ''A'' in declining order of arity is conventional but not required. The signature of an algebraic structure captures much of its essential nature apart from its axioms.
Example: an additive group over ''G'' has the signature ⟨''G'',+,-,0⟩ of type ⟨2,1,0⟩.
To allow for external operations, one considers structures whose universe is a union of several "sorts". (For example, a vector space may be conceived of as a 2-sorted algebra, with the two sorts "scalar" and "vector".) For each operation one has to prescribe which sorts are allowed as inputs, and which sort the output belongs to.

Contents

See also

References

See also

★ arity

★ universal algebra

★ signature (mathematical logic)

References

A monograph available free online:

★ Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''A Course in Universal Algebra.'' Springer-Verlag. ISBN 3-540-90578-2. Especially pp. 22-24.