Structure

Structure

A (mathematical) structure $\mathfrak{A}$ (for example, a group, field, vector space, set of complex numbers, etc.) can be represented as follows: $\mathfrak{A}=⟨A,R_i,F_j,C_k{⟩}_{i\in I,j\in J,k\in K}$ where: (i) $A$ is the universe under consideration (e.g., the set of group elements, the set of real numbers, etc.), and $A\ne \varnothing$ for technical reasons, (ii) each $R_i$ is an $n_i$-ary relation on $A$ for some natural number $n_i\ge 1$, i.e. $R_i\subseteq A^{n_i}$, (iii) each $F_j$ is an $n_j$-ary function on $A$ for some natural number $n_j\ge 1$, i.e. $F_j:A^{n_j}\to A$, (iv) each $C_k$ is a member of $A$, i.e. $C_k\in A$.

Remark

Many-Sorted Structure: In the case of a vector space, the universe could be the (disjoint) union of the set of scalars and the set of vectors. Unary relations $S,V$ could be used to single out these two sets/sorts. Relational Structure This is a structure in which there are only constants and relations. An $n$-ary function can be identified with an $(n+1)$-ary relation in the natural way.