Hypercontact semilattices

Contact Boolean algebras are one of the main algebraic tools in the region-based theory of space. T. Ivanova provided strong motivations for the study of mere semilattices with a contact relation. Another significant motivation for considering an even weaker underlying structure comes from event structures with binary conflict in the theory of concurrent systems in computer science. All the above-hinted notions deal with a binary contact relation. Several authors suggested the more general study of n-ary ‘hypercontact’ relations. A similar evolution occurred in the study of the just mentioned event structures in computer science. To unify the above lines of research, in this paper, we study joining semilattices with a hypercontact relation. We provide representation theorems into Boolean algebras. With a single exception, our proofs are choice-free. We also present several examples and problems; in particular, we briefly discuss some connections with event structures and hypergraphs.