Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Question

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$

Basically, this is saying that we may shuffle brackets right to get a larger $(\geq)$ expression, or left to get a smaller $(\leq)$ expression.

I'm looking for:

1. Interesting examples, especially those "from the wild."
2. The standard name for these kinds of structures, if such a name exists.

Near-example. Every distributive lattice satisfies the following. $$(x \vee y) \wedge z \leq x \vee (y \wedge z)$$

Motivation. In any such structure $(X,*)$ we can shuffle brackets around to get lower and upper bounds on an expression. For example, consider the following expression.

$$(x*y)*(x'*y')$$

We get the following lower and upper bounds, respectively.

$$((x*y)*x')*y',\quad x*(y*(x'*y'))$$

Answer 1

I came across this sub-associative property (that does seem like a good name) recently when I was thinking about ordered structures and enriched categories.

Let $(X,*)$ be a magma with a partial order $\leq$ such that we have $x*z\leq y*z$ and $z*x\leq z*y$ for all $z\in X$ whenever $x\leq y$ in $X$. I am assuming this is what you mean by a 'partially ordered magma'. Also suppose that there is a function $d\colon X\times X\to X$ satisfying $$\tag1 x*y\leq z\quad\Leftrightarrow\quad x\leq d(z,y) $$ for all $x,y,z\in X$. In other words, this condition says that for each $y\in X$ the multiplication function $-*y$ has an upper adjoint, which we denote $d(-,y)$. If the other multiplication function $y*-$ also had an upper adjoint then it would be fair to call $X$ a 'residuated magma', but let's not assume that as well.

I have written '$d$' for the upper adjoint of the multiplication function (this is non-standard notation) because we want to think of it as a very weak kind of "distance function". Specifically, we will assume that $d$ satisfies $$\tag2 d(x,y)*d(y,z)\leq d(x,z) $$ for all $x,y,z\in X$. This condition is motivated by the triangle inequality in a metric space and the composition morphism in an enriched category (this nLab page describes the connection between these two notions).

I now claim that $X$ has the sub-associative property.

Proof: Let $x,y,z\in X$. Since $x*(y*z)\leq x*(y*z)$ we have $x\leq d(x*(y*z),y*z)$ by (1), and similarly since $y*z\leq y*z$ we have $y\leq d(y*z,z)$ by (1). Now since $\leq$ is compatible with $*$ in the sense described above (i.e., since $X$ is a partially ordered magma) we have $$x*y\leq d(x*(y*z),y*z)*d(y*z,z),$$ and as such $x*y\leq d(x*(y*z),z)$ by (2). Finally, by (1) again we have $(x*y)*z\leq x*(y*z)$, as required.

An interesting feature of this set-up is that sub-associativity appears to be the best we can do. That is, even if we were to assume equality in (2), it seems impossible to derive the other sub-associativity inequality. Note, however, that if we had assumed that the other multiplication function also has an upper adjoint satisfying (2) then we would obtain $x*(y*z)\leq(x*y)*z$ for all $x,y,z\in X$ (and hence genuine associativity) by a similar proof.