Domain Range Semigroups and Finite Representations

Relational semigroups with domain and range are a useful tool for modelling nondeterministic programs. We prove that the representation class $R(D,R,*)$ is not finitely axiomatisable, answering arXiv:1811.01712 Question 5.1. We show that any signature containing $D, R, \smile, ;$ but not $-$ or $\cdot$ has the finite representation property, an extension of the result for ordered domain algebras $\{0,1,D,R, \leq, 1', \smile,;\}$. We survey the results in the area of the finite representation property and raise a number of open questions.