We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with in...

Non-classical logics, or variants of non-classical logics, have rapidly been developed together with the progress of computer science since the 20th century. Typically, we have found that many variants of non-classical logics are represented as ordered algebraic structures, more precisely as lattice expansions. From this point of view, we can think about the study of ordered algebraic structure...

Wilf stated that the Lagrange inversion formula (LIF) is a remarkable tool for solving certain kinds of functional equations, and at its best it can give explicit formulas where other approaches run into stone walls. Here we present the LIF combinatorially in the form of lattice paths, and apply it to the divisibility property of the coefficients of a formal power series expansion. For the LIF,...

The problem of imbedding an abstract distributive lattice in a Boolean algebra by an algebraic extension was suggested to the writer by M. H. Stone in 1933. Hausdorfff had already given a solution of this problem for the case where the given distributive lattice was a ring of point sets. A solution for the abstract case was presented by the writer to the Harvard Mathematical Colloquium (1934), ...

We generalize to universal algebra concepts originating from λ-calculus and programming to prove a new result on the lattice λT of λ-theories, and a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit a...

The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone repr...

Most applicable lattice reduction algorithms used in practice are BKZ (Block-Korkine-Zolotarev) type algorithms as the blockwise generalizations of the LLL algorithm (Lenstra-Lenstra-Lovasz). Its original version was proposed by Schnorr and Euchner in 1991. The quality of reduced lattice bases is measured by the Hermitian factor ||b1|| vol(L)1/d and the d-th root of this factor which is called ...

Modern humans possess a highly derived thumb that is robust and long relative to the other digits, with enhanced pollical musculature compared with extant apes. Researchers have hypothesized that this anatomy was initially selected for in early Homo in part to withstand high forces acting on the thumb during hard hammer percussion when producing stone tools. However, data are lacking on loads e...

BACKGROUND High ambient temperatures are a risk factor for nephrolithiasis, but the precise relationship between temperature and kidney stone presentation is unknown. OBJECTIVES Our objective was to estimate associations between mean daily temperature and kidney stone presentation according to lag time and temperatures. METHODS Using a time-series design and distributed lag nonlinear models...