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I only find the natural language description about a mathematical structure such as "a mathematical structure is a collection of objects together with some relations defined on the objects", or "a mathematical structure is a collection of objects together with some relations defined on the objects, and some axioms on the relations", etc. But I never find a strict definition of mathematical structure in terms of set(or other strictly defined mathematical concepts).

peter
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    That is a rigorous definition: a structure is a set with a (typically, finite) set of relations defined on it. – Conifold Sep 20 '22 at 03:53
  • So, a structure is a set, or two sets? Can you write the definition using math symbols? – peter Sep 20 '22 at 04:14
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    The structure $\mathcal{S}:=\langle S, R_1,\dots,R_N\rangle$, where $R_i\subset S^{n_i}$, i.e. $R_i$ are $n_i$-place relations on $S$. For example, a poset $\mathcal{S}$ is a set $S$ with a single binary ($2$-place) partial order relation $R$ on $S$ (typically written as $\leq$ instead of $R$). – Conifold Sep 20 '22 at 04:34
  • That is much clearer. Thank you, @Conifold! I'm wondering why when people talk about other math objects such as ordered pair, relation, or function, they can give rigorous definition, but when they talk about structure, they just describe it with natural language. – peter Sep 20 '22 at 06:19
  • BTW, the link for close of the question defines a natural number structure, which seems slightly different from your definition. The 0 seems not a relation. – peter Sep 20 '22 at 06:23
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    A definition is needed to prove things about entire classes of objects. But structures are not objects of ordinary mathematics, like pairs, relations, or functions, that one typically considers entire classes of. A structure is usually fixed and one works with objects within it, so a definition of what else it could be is redundant. Places where one would consider entire classes of structures and need a formal definition are mathematical logic or universal algebra, and those are niche subjects with a highly abstract flavor and narrow audience. – Conifold Sep 20 '22 at 07:16
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    See Mathematical structure and Structure (mathematical logic). – Mauro ALLEGRANZA Sep 20 '22 at 07:25
  • @Conifold Yes, recently, I read some materials about isomorphic sets in mathematical logic, which says a structure is a set together with an element for each constant symbol, a function together with each function symbol, and a relation for each relation symbol. This may be slightly different from your definition, but I think the difference is not important. What is important is that your tuple definition of structure lets me know what exactly a structure is. – peter Sep 21 '22 at 03:54
  • @MauroALLEGRANZA Thank you for providing me with the links. The definition in the second link is a little too complicated, especially, it has a recursive definition: a structure has a signature which is another structure. I think the Conifold's definition is enough for me at present. – peter Sep 21 '22 at 03:57

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