Topological Duality and Lattice Expansions, II: Lattice Expansions with Quasioperators

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators auS and a- on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals

Concurrent Kleene Algebra with Tests and Branching Automata

We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings , or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law (x‖y)(z‖w)≤(xz)‖(yw) of CKA, we make use of the subsumption order of Gischer on the gsp-strings. We also define deterministic branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests

Topological Duality and Lattice Expansions, I: A Topological Construction of Canonical Extensions

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone\u27s duality for distributive lattices rather than Priestley\u27s. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper

Some General Aspects of Exactness and Strong Exactness of Meets

Exact meets in a distributive lattice are the meets Λiai such that for all b, (Λiai) V b = Λi(ai V b); strongly exact meets in a frame are preserved by all frame homomorphisms. Finite meets are, trivially, (strongly) exact; this naturally leads to the concepts of exact resp. strongly exact filters closed under all exact resp. strongly exact meets. In [2], [12] it was shown that the subsets of all exact resp. strongly exact filters are sublocales of the frame of up-sets on a frame, hence frames themselves, and, somewhat surprisingly, that they are isomorphic to the useful frame Sc(L) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet-generated by open sublocales. In this paper we show that these are special instances of much more general facts. The latter concerns the free extension of join-semilattices to coframes; each coframe homomorphism lifting a general join-homomorphism φ : S --\u3e C (where S is a join-semilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (φ-precise filters) and a closure operator on C (and – a minor point – any closure operator on C is thus obtained). The former case is slightly more involved: here we have an extension of the concept of exactness (ψ-exactness) connected with the lifts of ψ: S --\u3e C with complemented values in more general distributive complete lattices C creating, again, frames of ψ-exact filters; as an application we learn that if such a C is join-generated (resp. meet-generated) by its complemented elements then it is a frame (resp. coframe) explaining, e.g., the coframe character of the lattice of sublocales, and the (seemingly paradoxical) embedding of the frame Sc(L) into it

Partially-Ordered Multi-Type Algebras, Display Calculi and the Category of Weakening Relations

We define partially-ordered multi-type algebras and use them as algebraic semantics for multi-type display calculi that have recently been developed for several logics, including dynamic epistemic logic [7], linear logic[10], lattice logic [11], bilattice logic [9] and semi-De Morgan logic [8]

Semi De Morgan Logic Properly Displayed

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi

Semi De Morgan Logic Properly Displayed

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi

Multi-type Display Calculus for Semi-De Morgan Logic

We introduce a proper multi-type display calculus for semi De Morgan logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Our proposal builds on an algebraic analysis of semi De Morgan algebras and applies the guidelines of the multi-type methodology in the design of display calculi

The Association of Depressive Symptoms With Brain Volume Is Stronger Among Diabetic Elderly Carriers of the Haptoglobin 1-1 Genotype Compared to Non-carriers

Aim: Depression is highly prevalent in type 2 diabetes and is associated with lower adherence to medical treatments, worse glycemic control, and increased risk for diabetes-related complications. The mechanisms underlying depression in type 2 diabetes are unclear. The haptoglobin (Hp) genotype is associated with type 2 diabetes related complications including increased risk for cerebrovascular pathology and worse cognitive performance. Its relationship with depression is unknown. We investigated the role of Hp genotype on the association of depression with brain and white matter hyperintensities (WMH) volumes.Methods: Depressive symptoms (measured with the 15-item Geriatric Depression Scale), brain MRI, and Hp genotypes, were examined in elderly subjects with type 2 diabetes [29 (13.8%) Hp 1–1 carriers and 181 (86.2%) non-carriers]. The interaction of Hp genotype with number of depressive symptoms on regional brain measures was assessed using regression analyses.Results: The significant interactions were such that in Hp 1–1 carriers but not in non-carriers, number of depressive symptoms was associated with overall frontal cortex (p = 0.01) and WMH (p = 0.04) volumes but not with middle temporal gyrus volume (p = 0.43).Conclusions: These results suggest that subjects with type 2 diabetes carrying the Hp 1–1 genotype may have higher susceptibility to depression in the context of white matter damage and frontal lobe atrophy. The mechanisms underlying depression in diabetes may differ by Hp genotype

opposition: “Model Theoretic Syntax focus[es]

The Call for Papers for this conference delineates the followin