We associate to a Hausdorff space, X, a double groupoid, ρ 2 (X), the homotopy double groupoid of X. The construction is based on the geometric notion of thin square. Under the equivalence of categories between small 2-categories and double categories with connection given in [BM] the homotopy double groupoid corresponds to the homotopy 2-groupoid, G2(X), constructed in [HKK]. The cubical natur...

We give a formulation of certain types of mechanical systems using the structure of groupoid of the tangent and cotangent bundles to the connguration manifold M; the set of units is the zero section identiied with the manifold M. We study the Legendre transformation on Lie algebroids. The purpose of this paper is not to prove new results but to investigate certain elementary tools utilized in A...

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that G is a principal groupoid. We show that C∗(G) is CCR if and only if G/G is a T1 topological space, and that C∗(G) is GCR if and only if G/G is a T0 topological space. We also show that C∗(G) is a Fell Algeb...

In [6–8] Murskii studied probability in algebra. He showed, for example, that the probability that a random groupoid is simple is 1 and that the probability that it has no idempotent element (one that satisfies a = a) is 1/e. To make these statements precise, define a groupoid table on the set {0, 1, . . . , n− 1} to be an n by n matrix with entries from {0, 1, . . . , n− 1}. Of course, each gr...

In this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid G(n). Also we show that G(n) contains commuting regular subsemigroup and give a necessary and sufficient condition for the groupoid G(n) to be commuting regular.

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the ...

We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1→ G00 → G → G/G00 → 1 where G00 is the “infinitesimal subgroup” of G and G/G00 is a compact real Lie group. We show that given a connected open subset U of G/G00 there is a canonical isomorphism between the fundamental group of U an...

X iv :0 80 3. 04 72 v1 [ m at h. R A ] 4 M ar 2 00 8 On decomposition of commutative Moufang groupoids B.V.Novikov It is well-known that the multiplicative groupoid of an alternative/Jordan algebra satisfies Moufang identities [1, 4]. Therefore it seems interesting to study the structure of such groupoids. In this note we apply to Moufang groupoids an approach which is widespread in Semigroup T...

We introduce the concept of vague ideals in a distributive implication groupoid and investigate their properties. The vague ideals of a distributive implication groupoid are also characterized.