Say I've got a structure $\mathfrak{M} := (M, \circ, \bot, \leq)$ with the following properties:
$\circ$ is total, associative, commutative and has neutral element $\bot$
$\leq$ is a linear order with minimum $\bot$, however, $\leq$ is unbound from above
$\circ$ respects the order itst. $a \leq a \circ b \wedge (a = a \circ b \iff b = \bot)$ f.a. $a, b$
I'm interested in the structure of the countable models for this theory. In order to get my head around this, I've split this up in several cases:
If $\leq$ is discrete, then many models are of the form $(X, \times^{\mathbb{N}}, 1^{\mathbb{N}}, \leq^{\mathbb{N}})$ where $X \subseteq \mathbb{N}$ is a (possibly infinitely generated) subset of natural numbers closed under the usual multiplication. Note that $(\mathbb{N}, +^{\mathbb{N}}, 0^{\mathbb{N}}, \leq^{\mathbb{N}})$ is isomorphic to the structure when $|X| = 1$ and $0 \not \in X$. So the "trivial" strucutre is handled.
If $\leq$ is dense, then my feeling is that I'm essentially dealing with $(Q, \times^{\mathbb{Q}}, 1^{\mathbb{Q}}, \leq^{\mathbb{Q}})$ with $Q := \{ q \in \mathbb{Q} \; | \; q \geq 1 \}$.
In the discrete case, my model could be a non-standard model, which would have the order type $\mathbb{N} + \mathbb{Q} \times \mathbb{Z}$, however, I'm not sure how many different interpretations of $\circ$ there are on this set respecting the axioms.
I'm not sure if there might be ordered sets that can function as a model with linear orders which are neither discrete nor dense, like $\mathbb{Q} \; \cap \; \{ (r, r + 1) \subset \mathbb{R} \; | \; r \in \mathbb{N} \} $ in which the multiplication would just round to the next nearest number. In this case, there would no longer be unique solutions, but I'm ok with that.
I'm not sure how to know which additional models there might be (non-standard companions for cases 2. and 4. probably). To me (a computer scientist), there might the be possibility that category theory or universal algebra are from help here, but I don't know any of these. So I didn't add the corresponding tags, feel free to do something about that if I'm mistaken.
So I'd like to know, if there an easy (read: computer-scientist-compatible) way to have a look at this (preferably something that is citable)?