On compositions of circular dense orders with structures and of their algebras of binary formulas

We consider compositions of structures and compositions of theories for circular dense orders and given structures, as well as related algebras. It is proved that for any $I$-groupoid $P$ consisting of non-negative labels, there is a theory $T$ with a complete type $p$ and a regular label function $\nu(p)$ such that the algebra of binary isolating formulas over the type $p$ is represented as a composition of a groupoid over a circular dense order and the groupoid $P$.