Within the framework of inquisitive semantics, we investigate the semantic prerequisites of an account of discourse coherence. In inquisitive semantics two views on meaning exist. Basic inquisitive semantics, InqB, follows from the view that to utter a sentence is to provide and request information (Roelofsen, 2011). Unrestricted inquisitive semantics, InqU, follows from the view that to utter a sentence is to propose to update the common ground in any of several ways (Ciardelli, Groenendijk, & Roelofsen, 2009). We illustrate with a simple example that InqU, but not InqB, can be a semantic foundation for an account of discourse coherence. However, the clauses of InqU have not been motivated conceptually with as much rigour as those of InqB, and they are technically not as well understood. In this paper we precise its conceptual motivation and, based on this conception of meaning, dene a semantics driven by general algebraic concerns. We show that the algebraic backbone of InqU is a commutative, idempotent semiring, which will facilitate an integration of inquisitive semantics with other formalisms. The algebraic structure gives rise to a compliance order on meanings that we put forward as a core notion for an account of discourse coherence.