Recent zbMATH articles in MSC 03G https://www.zbmath.org/atom/cc/03G 2021-04-16T16:22:00+00:00 Werkzeug On set-representable orthocomplemented difference lattices. https://www.zbmath.org/1456.06009 2021-04-16T16:22:00+00:00 "Su, Jianning" https://www.zbmath.org/authors/?q=ai:su.jianning An orthocomplemented difference lattice is an algebra $$\mathbf L=(L,\vee,\wedge,\Delta,{}',0,1)$$ of type $$(2,2,2,1,0,0)$$ such that $$(L,\vee,\wedge,{}',0,1)$$ is an orthocomplemented lattice and the following identities are satisfied: $$\Delta$$ is associative, $$x\Delta1\approx1\Delta x\approx x'$$ and $$x\Delta y\le x\vee y$$. $$\mathbf L$$ is called set-representable if there exists a set $$X$$ and a subset $$\Omega$$ of $$2^X$$ such that $$X\in\Omega$$, $$A\Delta B\in\Omega$$ for all $$A,B\in\Omega$$ and $$(L,\le,\Delta,0,1)\cong(\Omega,\subseteq,\Delta,\emptyset,X)$$. It is proved that the class of set-representable orthocomplemented difference lattices is not locally finite. An orthomodular lattice is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation and finite orthogonal joins given by disjoint unions. An orthomodular lattice $$\mathbf M$$ is said to be OML-embeddable into $$\mathbf L$$ if $$\mathbf M$$ can be embedded into the orthomodular lattice $$(L,\vee,\wedge,{}',0,1)$$. It is proved that not every concrete orthomodular lattice can be OML-embedded into an orthocomplemented difference lattice. Reviewer's remark: In the first reference Länger, H. M.'' should be Länger, H.''. Reviewer: Helmut Länger (Wien) A domain-theoretic investigation of posets of sub-$$\sigma$$-algebras (extended abstract). https://www.zbmath.org/1456.06011 2021-04-16T16:22:00+00:00 "Battenfeld, Ingo" https://www.zbmath.org/authors/?q=ai:battenfeld.ingo Summary: Given a measurable space $$(X,\mathcal{M})$$ there is a (Galois) connection between sub-$$\sigma$$-algebras of $$\mathcal{M}$$ and equivalence relations on $$X$$. On the other hand equivalence relations on $$X$$ are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-$$\sigma$$-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question. For the entire collection see [Zbl 1391.03010]. A categorical reconstruction of quantum theory. https://www.zbmath.org/1456.18014 2021-04-16T16:22:00+00:00 "Tull, Sean" https://www.zbmath.org/authors/?q=ai:tull.sean This paper lies in the long-standing tradition of \textit{reconstruction theorems}. To mention a notable few, we have \begin{itemize} \item The so-called \textit{Veblen-Young theorem} [\textit{O. Veblen} and \textit{J. W. Young}, Am. J. Math. 30, 347--380 (1908; JFM 39.0606.01); Bull. Sci. Math., II. Sér. 44, 105--112 (1920; JFM 47.0582.08); Projective Geometry. Vol. I. Boston and London: Ginn and Comp (1910; JFM 41.0606.06); Projective geometry. Vol. I, II. New York-Toronto-London: Blaisdell Publishing Company (1965; Zbl 0127.37604)] claims that a \textit{projective space} of dimension at least $3$ can be constructed as the projective space associated to a vector space over a division ring. Geometry of linear subspaces of a projective space or \textit{projective geometry} in short was axiomatized lattice-theoretically as finite-dimensional complemented modular lattices. \textit{G. Birkhoff} [Lattice theory. New York: American Mathematical Society (AMS) (1940; Zbl 0063.00402)] has shown that every complemented modular lattice of finite dimension is the direct product of lattices associated with projective geometries of finite dimension. \textit{J. von Neumann} [Proc. Natl. Acad. Sci. USA 22, 92--100 (1936; Zbl 0014.22307); Continuous geometry. Princeton, N.J.: Princeton University Press (1960; Zbl 0171.28003)] generalized these considerations to continuous geometry under the name of \textit{coordination theorems}. \item Quantum mechanics was formulated by orthomodular lattices. It was \textit{C. Piron} [Helv. Phys. Acta 37, 439--468 (1964; Zbl 0141.23204)] who succeeded in establishing the coordination theorem. See Theorem 7.44 in [\textit{V. S. Varadarajan}, Geometry of quantum theory. Vol. I. Princeton, N.J.-Toronto-London-Melbourne: D. van Nostrand Company, Inc. (1968; Zbl 0155.56802)] for its details. \item The category of modules was axiomatized categorically as \textit{abelian categories}. The so-called \textit{Freyd-Mitchell embedding theorem} [\textit{P. Freyd}, Abelian categories. An introduction to the theory of functors. New York-Evanston-London: Harper and Row, Publishers (1964; Zbl 0121.02103); \textit{B. Mitchell}, Theory of categories. New York and London: Academic Press (1965; Zbl 0136.00604)] claims that every abelian category is a full subcategory of a category of modules over some ring $R$ and that the embedding is an exact functor. \item Grothendieck toposes were axiomatized categorically as \textit{elementary toposes} by Lawvere and Tierney during the year 1969--1970. It was \textit{J. Giraud} [Cohomologie non abélienne. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0226.14011)] who succeeded in categorically characterizing elementary toposes for which one can reconstruct Grothendieck toposes. \end{itemize} This paper presents a genuinely categorical formalism of quantum mechanics based on dagger categories, establishing its coordination theorem to recover something like the standard Hilber-space formalism [\textit{D. Hilbert} et al., Math. Ann. 98, 1--30 (1927; JFM 53.0849.03)]. Reviewer: Hirokazu Nishimura (Tsukuba) Composition-nominative logics as institutions. https://www.zbmath.org/1456.03054 2021-04-16T16:22:00+00:00 "Chentsov, Alexey" https://www.zbmath.org/authors/?q=ai:chentsov.alexey-a "Nikitchenko, Mykola" https://www.zbmath.org/authors/?q=ai:nikitchenko.mykola-s Summary: Composition-nominative logics (CNL) are program-oriented logics. They are based on algebras of partial predicates which do not have fixed arity. The aim of this work is to present CNL as institutions. Homomorphisms of first-order CNL are introduced, satisfaction condition is proved. Relations with institutions for classical first-order logic are considered. Directions for further investigation are outlined. On a generalization of equilogical spaces. https://www.zbmath.org/1456.03110 2021-04-16T16:22:00+00:00 "Pasquali, Fabio" https://www.zbmath.org/authors/?q=ai:pasquali.fabio Summary: We use the theory of triposes to prove that every (non-degenerate) locale $$\mathsf{H}$$ is the set of truth values of a complete and co-complete quasi-topos into which the category of topological spaces embeds and the topos of sheaves over $$\mathsf{H}$$ reflectively embeds. Semantics of a typed algebraic lambda-calculus. https://www.zbmath.org/1456.03029 2021-04-16T16:22:00+00:00 "Valiron, Benoît" https://www.zbmath.org/authors/?q=ai:valiron.benoit Summary: Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations. For the entire collection see [Zbl 1445.68010]. Realizability in ordered combinatory algebras with adjunction. https://www.zbmath.org/1456.03025 2021-04-16T16:22:00+00:00 "Ferrer Santos, Walter" https://www.zbmath.org/authors/?q=ai:ferrer-santos.walter-ricardo "Guillermo, Mauricio" https://www.zbmath.org/authors/?q=ai:guillermo.mauricio "Malherbe, Octavio" https://www.zbmath.org/authors/?q=ai:malherbe.octavio Summary: In this work, we continue our consideration of the constructions presented in the paper \textit{Krivine's Classical Realizability from a Categorical Perspective} by \textit{T. Streicher} [Math. Struct. Comput. Sci. 23, No. 6, 1234--1256 (2013; Zbl 1326.03083)]. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented, \textit{mutatis mutandis}, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.