2,324 research outputs found
On the coset category of a skew lattice
Skew lattices are non-commutative generalizations of lattices. The coset
structure decomposition is an original approach to the study of these algebras
describing the relation between its rectangular classes. In this paper we will
look at the category determined by these rectangular algebras and the morphisms
between them, showing that not all skew lattices can determine such a category.
Furthermore, we will present a class of examples of skew lattices in rings that
are not strictly categorical, and present sufficient conditions for skew
lattices of matrices in rings to constitute -distributive skew
lattices.Comment: 17 pages, submitted to Demonstratio Mathematica. arXiv admin note:
text overlap with arXiv:1212.649
Thin Games with Symmetry and Concurrent Hyland-Ong Games
We build a cartesian closed category, called Cho, based on event structures.
It allows an interpretation of higher-order stateful concurrent programs that
is refined and precise: on the one hand it is conservative with respect to
standard Hyland-Ong games when interpreting purely functional programs as
innocent strategies, while on the other hand it is much more expressive. The
interpretation of programs constructs compositionally a representation of their
execution that exhibits causal dependencies and remembers the points of
non-deterministic branching.The construction is in two stages. First, we build
a compact closed category Tcg. It is a variant of Rideau and Winskel's category
CG, with the difference that games and strategies in Tcg are equipped with
symmetry to express that certain events are essentially the same. This is
analogous to the underlying category of AJM games enriching simple games with
an equivalence relations on plays. Building on this category, we construct the
cartesian closed category Cho as having as objects the standard arenas of
Hyland-Ong games, with strategies, represented by certain events structures,
playing on games with symmetry obtained as expanded forms of these arenas.To
illustrate and give an operational light on these constructions, we interpret
(a close variant of) Idealized Parallel Algol in Cho
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