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We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with… (More)

We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi-groups and dynamic predicate logic.

The sequential calculus of von Karger and Hoare [18] is designed for reasoning about sequential phenomena, dynamic or temporal logic, and concurrent or reactive systems. Unlike the classical calculus of relations, it has no operation for forming the converse of a relation. Sequential algebras [15] are algebras that satisfy certain equations in the… (More)

This note gives an explicit construction of the one-generated free domain semiring. In particular it is proved that the elements can be represented uniquely by finite antichains in the poset of finite strictly decreasing sequences of nonnegative integers. It is also shown that this domain semiring can be represented by sets of binary relations with union,… (More)

In the present paper, which is a sequel to [20, 4, 12], we investigate further the structure theory of quasi-MV algebras and √ quasi-MV algebras. In particular: we provide a new representation of arbitrary √ qMV algebras in terms of √ qMV algebras arising out of their MV* term sub-reducts of regular elements; we investigate in greater detail the structure… (More)

Concurrent Kleene algebras were introduced by Hoare, Möl-ler, Struth and Wehrman in [HMSW09,HMSW09a,HMSW11] as idem-potent bisemirings that satisfy a concurrency inequation and have a Kleene-star for both sequential and concurrent composition. Kleene algebra with tests (KAT) were dened earlier by Kozen and Smith [KS97]. Concurrent Kleene algebras with tests… (More)