How to differentiate structures with the same operations but in a different order?

Question

In model theory, an algebraic structure is an ordered pair consisting of a set $S$ as its first component and a set of finitary operations on $S$ as its second component. However, this definition does not take note of the order of operations in that structure. For example, a field $(F;+,*)$ is not a field under the reversed order $(F;*,+)$. And also, sometimes we want to repeat an operation two or more times, while bare sets do not have repetition. So, can someone clear up my confusion and give me a rigorous and precise clarification of this issue?

Answer 1

In model theory, an algebraic structure is an ordered pair consisting of a set $S$ as its first component and a set of finitary operations on $S$ as its second component.

That's not really correct. In model theory we have the notion of a $\Sigma$-structure for a language $\Sigma$; a $\Sigma$-structure is a pair $(S,\mathfrak{I})$ where $S$ is a (usually nonempty) set and $\mathfrak{I}$ is a function with domain $\Sigma$. To be precise, an interpretation of a language $\Sigma$ on a set $S$ is a function $$\mathfrak{I}:\Sigma\rightarrow S\sqcup(\bigcup_{n\in\omega}\mathcal{P}(S^n))\sqcup(\bigcup_{n\in\omega}Func(S^n,S))$$ (where $Fun(A,B)$ is the set of functions from $A$ to $B$) such that $\mathfrak{I}$ sends each constant symbol to an element of $S$, each $n$-ary relation symbol to an element of $\mathcal{P}(S^n)$, and each $n$-ary function symbol to an element of $Func(S^n,S)$.

The functional nature of the interpretation, while often suppressed, is crucial at a technical level. Even if $\Sigma$ consists only of function symbols, $\mathfrak{I}$ is more than just a set of functions on $S$ - it also indicates which function corresponds to which symbol in the language $\Sigma$.

For example, and using red to indicate symbols as opposed to actual functions/relations/constants (= their interpretations), we're looking at the language $\Sigma=\{\color{red}{\oplus},\color{red}{\otimes}\}$ consisting of two binary function symbols. Given a set $F$ and binary functions $f,g:F^2\rightarrow F$ on $F$, we can then consider two distinct interpretations which use the functions $f$ and $g$: $$\mathfrak{I}_1: \color{red}{\oplus}\mapsto f, \color{red}{\otimes}\mapsto g\quad\mbox{and}\quad\mathfrak{I}_2:\color{red}{\oplus}\mapsto g, \color{red}{\otimes}\mapsto f.$$ The pairs $(F,\mathfrak{I}_1)$ and $(F,\mathfrak{I}_2)$ are manifestly different objects, and this reflects the "change in order" of the functions involved.

Similarly, possible repetition of functions(/relations/constants) corresponds to non-injectivity of the interpretation involved (e.g. $\mathfrak{I}_3: \color{red}{\oplus},\color{red}{\otimes}\mapsto f$), which is completely unproblematic.