Koskas,

Groupoides, demi-hypergroupes et hypergroupes, J.

Now, we have two PBW-basis of [B.sub.q] (and correspondingly of [B.sub.q]), namely Kharchenko's PBW-basis and the PBW-basis defined from a reduced expression of a longest element of the Weyl

groupoid. But both basis are reconciled by [AY, Theorem 4.12], thanks to [A2, 2.14].

A left almost semigroup S is a mid structure between a

groupoid and a commutative semigroup.

A neutrosophic

groupoid satisfying the left invertive law is called a neutrosophic left almost semigroup and is abbreviated as neutrosophic LA-semigroup.

A Lie

groupoid is a

groupoid G whose set of arrows and set of objects are both manifolds whose structure maps s, t, e, i, m are all smooth maps and s, t are submersions.

(ii) An m-ary

groupoid (R, f) is called an m-ary semi-group if f is associative (from Dudek [24]); that is, if

A Smarandache

groupoid G is a

groupoid which has a proper subset S [subset] G which is a semi-group under the operation of G.

This operation over which the various 'directions' are taken (1) subsequently determines the holonomy of the system through an error-correction network--a broader scale geometric representation of transitional phases in which the broken symmetries may be expressed in terms of holonomy groups that collectively, via disjoint union, form a holonomy

groupoid, a structure which in principle can be given explicitly.

A subset I of an right modular

groupoid S is called left (right) ideal of S if SI [subset or equal to] I(IS [subset or equal to] I).

NIS TO R, Analysis ofgeometric operators on open manifolds: a

groupoid approach, in Quantization of Singular Symplectic Quotients, N.

for all u, v, w [member of] [R.sup.3.sub.c], and they turn out to be automorphisms of the Einstein

groupoid ([R.sup.3.sub.c], [direct sum]).

Heller's current work focuses on the fields of noncommutative geometry and

groupoid theory in mathematics.