190 research outputs found
Supertropical semirings and supervaluations
We interpret a valuation on a ring as a map into a so
called bipotent semiring (the usual max-plus setting), and then define a
\textbf{supervaluation} as a suitable map into a supertropical semiring
with ghost ideal (cf. [IR1], [IR2]) covering via the ghost map . The set \Cov(v) of all supervaluations covering has a natural
ordering which makes it a complete lattice. In the case that is a field,
hence for a Krull valuation, we give a complete explicit description of
\Cov(v).
The theory of supertropical semirings and supervaluations aims for an algebra
fitting the needs of tropical geometry better than the usual max-plus setting.
We illustrate this by giving a supertropical version of Kapranov's lemma.Comment: 47 page
Tensor products and regularity properties of Cuntz semigroups
The Cuntz semigroup of a C*-algebra is an important invariant in the
structure and classification theory of C*-algebras. It captures more
information than K-theory but is often more delicate to handle. We
systematically study the lattice and category theoretic aspects of Cuntz
semigroups.
Given a C*-algebra , its (concrete) Cuntz semigroup is an object
in the category of (abstract) Cuntz semigroups, as introduced by Coward,
Elliott and Ivanescu. To clarify the distinction between concrete and abstract
Cuntz semigroups, we will call the latter -semigroups.
We establish the existence of tensor products in the category and study
the basic properties of this construction. We show that is a symmetric,
monoidal category and relate with for
certain classes of C*-algebras.
As a main tool for our approach we introduce the category of
pre-completed Cuntz semigroups. We show that is a full, reflective
subcategory of . One can then easily deduce properties of from
respective properties of , e.g. the existence of tensor products and
inductive limits. The advantage is that constructions in are much easier
since the objects are purely algebraic.
We also develop a theory of -semirings and their semimodules. The Cuntz
semigroup of a strongly self-absorbing C*-algebra has a natural product giving
it the structure of a -semiring. We give explicit characterizations of
-semimodules over such -semirings. For instance, we show that a
-semigroup tensorially absorbs the -semiring of the Jiang-Su
algebra if and only if is almost unperforated and almost divisible, thus
establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results
corrected, in particular added 5.2.3-5.2.
Radicals and Ideals of Affine Near-semirings over Brandt Semigroups
This work obtains all the right ideals, radicals, congruences and ideals of
the affine near-semirings over Brandt semigroups.Comment: In Proceedings of the International Conference on Semigroups,
Algebras and Operator Theory (ICSAOT-2014), Kochi, Indi
Free Kleene algebras with domain
First we identify the free algebras of the class of algebras of binary
relations equipped with the composition and domain operations. Elements of the
free algebras are pointed labelled finite rooted trees. Then we extend to the
analogous case when the signature includes all the Kleene algebra with domain
operations; that is, we add union and reflexive transitive closure to the
signature. In this second case, elements of the free algebras are 'regular'
sets of the trees of the first case. As a corollary, the axioms of domain
semirings provide a finite quasiequational axiomatisation of the equational
theory of algebras of binary relations for the intermediate signature of
composition, union, and domain. Next we note that our regular sets of trees are
not closed under complement, but prove that they are closed under intersection.
Finally, we prove that under relational semantics the equational validities of
Kleene algebras with domain form a decidable set.Comment: 22 pages. Some proofs expande
Domain and Antidomain Semigroups
Abstract. 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.
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Automated verification of refinement laws
Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs
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