Multirelational representation theorems for complete idempotent left semirings

• Multirelational representation theorems for complete IL-semirings have been shown.
• Notions of powerset IL-semiring and invertibility for complete IL-semirings are introduced.
• Using invertibility, CCPA complete IL-semirings and powerset IL-semirings can be characterised.

Complete idempotent left semirings are a relaxation of quantales by giving up strictness and distributivity of composition over arbitrary joins from the left. It is known that the set of up-closed multirelations over a set forms a complete idempotent left semiring together with union, multirelational composition, the empty multirelation, and the membership relation. This paper provides a sufficient condition for a complete idempotent left semiring to be isomorphic to a complete idempotent left semiring consisting of up-closed multirelations, in which all joins, the least element, multiplication, and the unit element are respectively given by unions, empty multirelations, the multirelational composition, and the membership relation. Some equivalent conditions of the sufficient condition are also provided.