Let $R\subset V\times V$ be a relation on a set $V$. For a subset $S\subset V$, define its orthogonal complement with respect to $R$ as $$S^l:=\{ x: \forall y\in S\ \ (x,y)\in R\},\ \ S^r:=\{y: \forall x\in S\ \ (x,y)\in R\},\ S^{lr}:=(S^l)^r,...$$
For two words $u,v$ in alphabet $\{l,r\}$, write $u =_{V,R,S} v$ iff $S^u=S^v$. Thus, $l=_{V,R,S} lrl$ and $r=_{V,R,S} rlr$.
This is the usual orthogonal complement when $V$ is a vector space and $R:=\{(u,v):\phi(x,y)=0\}$ for a bilinear form $\phi(-,-)$.
Fix $R$, and define a semi-group $O_{V,R}:=<l,r: u=v\ \text{ if }\ \forall S\subset V\ u=_{V,R,S}v >$ with two generators $l,r$ and relations $u=v$ whenever $\forall S\subset V\ u=_{R,S}v$. The relations $l=_{V,R,S} lrl$ and $r=_{V,R,S} rlr$ seem to hold only for the two generators, and in particular, it seems these semi-groups are not necessarily regular.
What is known about the regular semigroups of this form ? Have they been studied ? Is there a name ?
I am particularly interested in the case when $V$ is the set of morphisms of a category, and $R$ is the weak ortogonality relation, also known as the lifting property.
The simplest non-trivial case of interest is as follows: $X$ is the set of continuous maps in the category of topological spaces, and $R$ is the weak orthogonality relation, and $S:=\{ \emptyset\to \{pt\}\}$ is the class consisting of a single map from the empty set to a singleton.
Is something known when $V$ is a vector space and $R$ is a possibly non-symmetric bilinear form ? (I thank @BenjaminSteinberg for corrections).
