In the transition from set theory to order theory, what is the appropriate generalization of "group"?

Question

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of the symmetric group on its underlying set.

Question. If we want to replace "set" with "poset," and "bijective endofunction" with "endomorphism that occurs as the lower adjoint of some Galois connection," what is the correct concept to replace "group" with?

Answer 1

This question is considered in the book Lattices and Ordered Algebraic Structures by Blyth. Given a poset $X$, he writes $\operatorname{Res}(X)$ for the semigroup of 'residuated' functions (i.e., lower adjoints) $X\to X$, and presents the following result of Johnson.

Theorem 1.12: If $X$ is a bounded poset then $\operatorname{Res}(X)$ is a generalised Baer semigroup.

Generalised Baer semigroups (or 'pre-Baer' semigroups in Johnson's terminology) are characterised by some annihilator conditions. Specifically, a semigroup $S$ having an identity element $1\in S$ and a zero element $0\in S$ is a generalised Baer semigroup if for each $a\in S$ there exist $a_r,a_l\in S$ satisfying $LR(a)=L(a_r)$ and $RL(a)=R(a_l)$, where $$\begin{align}R(b)&=\{c\in S:bc=0\}\quad\text{and}\\ L(b)&=\{c\in S:cb=0\}\end{align}$$ are the right and left annihilators of $b\in S$.