| Signature (universal Algebra) |
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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 s, 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 Axiom s. Example: an Additive Group over ''G'' has the signature ⟨''G'',+,-,0⟩ of type ⟨2,1,0⟩. To allow for External Operation s, 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. SEE ALSO REFERENCES A monograph available free online:
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