Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
432979 | Journal of Logical and Algebraic Methods in Programming | 2015 | 16 Pages |
•Symmetric Heyting relation algebras (SHRAs) generalize relation algebras.•SHRAs have an adjoint pair of converse operations.•Within an SHRA we construct the sub-SHRA induced by a pre-order element.•Relations on a hypergraph form an SHRA and arise from this construction.•Results are applied to mathematical morphology on hypergraphs.
A relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting algebra that supports an order-reversing involution. A general construction for these symmetric Heyting relation algebras is given which includes as a special case the algebra of relations on a hypergraph. A particular feature of symmetric Heyting relation algebras is that instead of an involutory converse operation they possess both a left converse and a right converse which form an adjoint pair of operations. Properties of the converses are established and used to derive a generalization of the well-known connection between converse, complement, erosion and dilation in mathematical morphology. This provides part of the foundation necessary to develop mathematical morphology on hypergraphs based on relations on hypergraphs.