This is basically a reference request, but the post is going to be relatively long (and a little bit verbose): I apologize in advance for that.
Premise. There are several examples of "ordered structures" appearing in nature (at least for the moment, let me be vague with the actual meaning of various terms I'm using): ordered groups, ordered rings, ordered vector spaces, etc. All of these have something in common: They are algebraic structures, where each operation is compatible, in some sense and in some way, with a (partial) order. Thus, I find it reasonable to ask:
What about a fully formal definition of an ordered structure, in general?
For simplicity, I'm not seeking a definition worded in the language of categories, and I'm looking at the case where "structure" means whatever can be understood as a (classical) model of a finitary, single-sorted, first-order algebraic theory in the frame of model theory (in principle, I'm also interested in possibly non-finitary or many-sorted structures, but the general case would probably make things look more complicated than they are and overshadow some relevant details).
The most basic, non-trivial example is provided by structures with one operation. For instance, an ordered semigroup is a triple $\mathbb A = (A, {\cdot}\,, \preceq)$, where $(A, \cdot)$ is a semigroup (written multiplicatively) and $\preceq$ an order on $A$ with the property that $xz \preceq yz$ and $zx \preceq zy$ for all $x,y,z \in A$ such that $x \preceq y$.
Of course, the axiom of associativity doesn't play any role in these definitions, so from a fundamental point of view it would have been morally better to replace "semigroups" with "magmas" in the previous paragraph, but on the other hand, that doesn't really matter, insofar as I'm just using ordered semigroups as a guiding example to forge a tentative answer to my question along the following lines.
Naive (and "wrong") approach. Let $T$ be a finitary, single-sorted, first-order theory, which, to me, means a triple $(\sigma, \Xi, V)$, where:
- $V$ is an infinite countable set of (logical) variables from the formal language, $\mathcal L$, underlying the axiomatic theory used to lay out the foundations (say, Tarski-Grothendieck set theory, as oversized as it may appear);
- $\sigma$ is a (single-sorted) signature, namely a triple $(\Sigma_{\rm f}, \Sigma_{\rm r}, \varrho)$ consisting of a set of function symbols $\Sigma_{\rm f}$, a set of relation symbols $\Sigma_{\rm r}$, and a function $\varrho: \Sigma_{\rm f} \cup \Sigma_{\rm r} \to \mathbf N^+$ such that $\Sigma_{\rm f}$, $\Sigma_{\rm r}$, and the set of (logical and non-logical) symbols of $\mathcal L$ are pairwise disjoints (the function $\varrho$ assigns to each function or relation symbol of $\sigma$ an ariety; in particular, an $n$-ary function symbol will correspond, in any model of the theory, to a function $A^{n-1} \to A$ (for some set $A$), and an $n$-ary relation symbol to a subset of $A^n$);
- $\Xi$ is a (possibly empty) subset of $\langle V; \sigma \rangle$, the set of all (well-formed) formulas generated by combining, according to the formation rules of first-order logic, the variables in $V$, the (function and relation) symbols of $\sigma$, and those of $\mathcal L$ which are not logical variables.
(The role of $V$ is meaningless here, as we could assume that $V$ includes all the variables in $\mathcal L$; however, it starts being meaningful in the case of many-sorted theories, and that's why I beg your indulgence for being redundant in this respect.)
Now, $T$ is called an algebraic theory if $\Sigma_{\rm r}$ is empty (in which case $\sigma$ is referred to as an algebraic signature), and since we are interested in structures that are algebraic except for the fact of being endowed with an order, we may refer to $T$ as a quasi-algebraic theory if
- $\Sigma_{\rm r}$ consists of one relation symbol $R$, subjected to the axioms of reflexivity, antisymmetry, and transitivity (for binary relations), which are hence comprised among the formulas in $\Xi$ (so that $R$ is interpreted as a partial order in any possible model of $T$);
- $\Xi$ includes basic formulas encoding the compatibility between each function symbol of $\sigma$ and the relation $R$, which can be phrased as follows: If $\varsigma$ is a function symbol in $\Sigma_{\rm f}$ of ariety $n := \varrho(\varsigma)$, then the formula
$$
\forall \vec{x}, \vec{y} \in V^{n-1}: R^{(2n-2)}(\vec{x}, \vec{y}) \implies R(\varsigma(\vec{x}), \varsigma(\vec{y}))\qquad\qquad(\star)
$$
belongs to $\Xi$. (The notation should be self-explanatory, but let me note that the formula in the case of constant symbols is redundant, in the sense that it is implied, in any model of theory, by the very fact that $R$ is reflexive; yet, having it there makes the approach look more "uniform" than otherwise, in the same spirit of Joel David Hamkins' comment below.)
An ordered structure would then be any model of a quasi-algebraic theory.
Unfortunately, the above doesn't work even in the case of (the theory of) groups, where $\Sigma_{\rm f}$ consists, say, of the symbols $\cdot$ (binary symbol for "multiplication"), ${}^{-1}$ (unary symbol for "inversion"), and $1$ (constant symbol for "identity"). In fact, the naive approach would suggest to assume one binary relation symbol, say $\preceq$, in $\Sigma_{\rm r}$ and the following axioms in $\Xi$:
$$
\begin{split}
(1)\ & \forall x, y, z \in V: x \cdot (y \cdot z) = (x \cdot y) \cdot z \\
(2)\ & \forall x \in V: x \cdot 1 = 1 \cdot x = x \\
(3)\ & \forall x,y \in V: x \cdot x^{-1} = x^{-1} \cdot x = 1 \\
(4)\ & \forall x,y,z \in V: (x \preceq x) \land (((x \preceq y) \land (y \preceq y) \implies (x=y))) \land (((x \preceq y) \land (y \preceq z) \implies (x \preceq z))) \\
(5)\ & \forall x,y \in V: (x \preceq y) \implies (x^{-1} \preceq y^{-1}) \\
(6)\ & \forall x,y,z \in V: (x \preceq y) \implies ((x \cdot z \preceq y \cdot z) \land (z \cdot x \preceq z \cdot y)) \\
\end{split}
$$
The problem is that the 5th axiom is essentially incompatible with the 6th, as the latter yields that, in every model $(A; +, \cdot\,,1;\preceq)$ of $T$, we have $x \preceq y$ iff $y^{-1} \preceq x^{-1}$.
Remedies. One may argue that the "mistake" lies in the fact of including the function symbol ${}^{-1}$ in the signature of the theory, while an alternative could be to avoid it and replace the 3th axiom above with:
$$
\forall x \in V, \exists\, \tilde{x} \in V: x \cdot \tilde{x} = \tilde{x} \cdot x = 1.
$$
This is certainly a possibility (as long as $\mathcal L$ includes both $\forall$ and $\exists$ among the logical symbols), but I find it rather "artificial" (whatever it may mean), all the more that the same strategy fails if we try to use the naive approach outlined in the above to recover as a special case the common definition of an ordered ring, for which the compatibility between multiplication and order is "restricted to nonnegative factors", which has no hope to fit in the paradigm implied by condition $(\star)$.
So, putting it all together, my (second) question is:
Where should I look up for a sufficiently general definition of "ordered structure", which copes with the kind of issues that I've tried to point out in this post?
I've my own ideas on what to do, but there may be a much better way on how to proceed, which is what I'm looking for. In particular, one solution could be as follows:
- Start with a finitary, single-sorted, first-order theory $T = (\sigma, \Xi, V)$, whose signature is of the form $((\varsigma_i)_{i \in I}, (R_i)_{i \in I \,\cup\, \{\infty\}}, \varrho)$ for some index set $I$ and $\infty \notin I$, so that each function symbol $\varsigma$ has one corresponding relation symbol of ariety $2\varrho(\varsigma)-2$, and we have an extra relation symbol $R_\infty$.
- Make $\Xi$ include, for each relation symbol $R$ of $\sigma$, the axioms of reflexivity, antisymmetry, and transitivity (extended, as appropriate, from binary to $2n$-ary relations), so that, in particular, $R_\infty$ is interpreted as an
order in any possible model of the theory. - Rewrite condition ($\star$) so as to replace, for each
function symbol $\varsigma$ of $\sigma$, the symbol "$R^{(2n-2)}$" to the left of the connective "$\implies$" with the (unique) relation
corresponding to the function symbol $\varsigma$, and the symbol "$R$" to the right with "$R_\infty$".
These conditions together describe a paradigm I will refer to as (P), the last condition encoding the naive idea that the relations $R_i$ must be glued together in a "consistent way" (and can't be "completely independent" from each other, which otherwise would result into something exceedingly general, I feel).
An example. Assume the (algebraic) theory of unital rings is encoded by the (algebraic) signature whose set of function symbols is given by $\Sigma_{\rm ring} := \{+, \cdot\,, -\,, 0, 1\}$, where $+$ and $\cdot$ are, respectively, the binary symbols for "addition" and "multiplication", $-$ is the unary symbol for "additive inverse", and $0$ and $1$ are, respectively, the constant symbols for "additive identity" and "multiplicative identity". The axioms of the theory are the usual ones:
$$
\begin{split}
(1)\ & \forall x, y, z \in V: (x+(y+z) = (x+y)+z) \land (x \cdot (y \cdot z) = (x \cdot y) \cdot z) \\
(2)\ & \forall x \in V: (x+0 = 0 + x = x) \land (x \cdot 1 = 1 \cdot x = x) \\
(3)\ & \forall x,y \in V: x + (-x) = (-x) + x = 0 \\
(4)\ & \forall x,y,z \in V: (x \cdot (y+z) = (x \cdot y) + (x \cdot z)) \land ((y+z) \cdot x = (y \cdot x) + (z \cdot x)) \\
\end{split}
$$
Now, according to the paradigm (P), the theory of ordered rings would have a signature of the form $(\Sigma_{\rm ring}, \Sigma_{\rm r}, \varrho)$, where $\Sigma_{\rm r}$ is a set of relation symbols of the form $\{R_{(+)}, R_{(\cdot)}, R_{(-)}, R_{(0)}, R_{(1)}, \preceq\}$, with each relation symbol subjected to the axioms of reflexivity, antisymmetry, and transitivity, all the relation symbols glued together by the 4th condition of (P), and each function symbol $\varsigma$ in $\Sigma_{\rm ring}$ "bound" to the corresponding relation symbol $R_{(\varsigma)} \in \Sigma_{\rm r}$ by the formula:
$$
\forall (\vec{x}, \vec{y}) \in V^{n-1} \times V^{n-1}: R_{(\varsigma)}(\vec{x}, \vec{y}) \implies (\varsigma(\vec{x}) \preceq \varsigma(\vec{y})),
$$
where $n$ is the ariety of $\varsigma$.
In particular, this paradigm fits with the usual notion of an ordered ring, which is then a tuple $\mathbb A = (A; +, \cdot\,,-\,,0,1; R_{(+)}, R_{(\cdot)}, R_{(-)}, \preceq)$, where $\preceq$ is a partial order on $A$, $R_{(+)}$ the subset of $A^2 \times A^2$ consisting of those pairs $((x,y),(x,z))$ or $((y,x),(z,x))$ with $y \preceq z$, $R_{(\cdot)}$ the subset of $A^2 \times A^2$ consisting of those pairs $((x,y),(x,z))$ or $((y,x),(z,x))$ such that $0 \preceq x$ and $y \preceq z$, and $R_{(-)} = \{(x,x): x \in A\}$. (I'm intentionally omitting any explicit reference in models to the relations associated with constants, for they don't add any information, as a result of considerations already made in the above.)
Best Answer
You might be interested in a universal algebraic perspective.
There is a concept called polarity based on the relation "algebraic operation f preserves or respects n-ary relation (subuniverse) R". If you have picked a partial, pre, quasi, or total order R, you might consider studying the structure with all functions preserving that order. Similarly, given an operation f, consider all subuniverses of product of the algebra which f respects, and see which of those are the kind you wish to have as an order. If preservation is important to you, you might consider this type of approach.
In my limited experience, an ordered algebraic structure is an algebraic structure equipped with a partial order such that there is an interesting interplay that results in useful theorems about such structures. Often one or more of the operations preserve the order, but not all of them, as you have observed. I suspect your attempt at generalization will be more successful if you have a target theorem, to make up an example: "an algebraic structure can be equipped with an (interesting) partial order iff the algebra generates a congruence permutable variety". If you want examples of target theorems, you might browse the journal 'Order' for some inspiration.
Gerhard "What Uses Are You Intending?" Paseman, 2015.12.24