Normal design algebra

We generalise the designs of Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, e.g., forming a Kleene and omega algebra of designs. Moreover, we prove a generalised fixpoint theorem for isotone functions on designs. We apply our framework to investigate symmetric linear recursion and its relation to tail-recursion; this substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, pre-image, convergence and noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP