The definitions of structure that I've read define it "a set containing mathematical objects endowed with an operation that enriches its structure." The inclusion of the word being defined in the definition is confusing. I've read several articles on structure - and as far as I can tell, it just denotes 'a set containing mathematical objects and some operations'. Is my definition of structure adequate?
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2It means that an (algebraic) structure is a set enriched by specified operations on the set.. A relational structure may also include relations too. – Bill Dubuque Aug 13 '17 at 17:04
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Your definition seems good for the algebraic structures, but note that we can have also more sets and more operations, as in the case of vector spaces or of modules.
Also, we can have ''structures'' defined by means of a relation (not an operation), as an ordered set, or adding to a set ( or to an underlying structure) some extra element, as a family of subsets with special properties (as in the case of topological or metric spaces) or a function with suitable properties (as a metric space or a manifold) .
Emilio Novati
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What about for example metrics/topologies/(order)relations? Those aren't operations so I don't think his definition is adequate. I know the question was tagged as abstract algebra but the question itself doesn't specify it – Jens Renders Aug 13 '17 at 15:19
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Emilio also gave other important examples of structures. In general I don't think one can say more than: a structure is a tuple of sets. Some of these sets may be "underlying sets", some functions and others relations (functions and relations are somewhat interchangable anyway).
However, to understand what really "makes a kind of structure work" you need to ask yourself what the homomorphisms of this kind of structure are supposed to look like. This will reflect what you really care about in your structure.
Stefan Perko
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