We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that VB-groupoids, extensively studied in the recen...

Groupoids satisfying the equation x(yz) = (xy)(xz) are called left distributive, or LD-groupoids. We give an algorithm for their enumeration and prove several results on the collection of LD-groupoids extending a given monounary algebra.

Slim groupoids are groupoids satisfying x(yz) ≈ xz. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such g...

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The g...

We approach Mackenzie’s LA-groupoids from a supergeometric point of view by introducing Q-groupoids. A Q-groupoid is a groupoid object in the category of Q-manifolds, and there is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. Using this approach, we associate to every LA-groupoid a double complex whose cohomology simultaneously generalizes Lie groupoid coh...

In this work we deal with coverings and actions of Lie groupgroupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth coverings of Lie group-groupoids.

Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic (T2CA-groupoids) are two types of non-associative which satisfy law type-2 law, respectively. In this paper, we prove theorems that weak cancellativity is right quasi-cancellativity left in a CA-groupoid, thus successfully solving open problems. Moreover, the relationships among separativity, commutativity CA-groupoid discussed. Fina...

The quasivariety of groupoids [Formula: see text] satisfying the implication [Formula: see text] generalises rectangular semigroups and central groupoids. We call them rectangular groupoids and find three combinatorial structures based upon arrays, matrices and graphs that are closely related. These generalise several groupoids of independent interest. The quasivariety generates the variety of ...

This paper deals with Lie groupoids, in particular symplectic and contact groupoids. We formulate a deenition of contact groupoids; we give examples and a counterexample (groupoid with a contact form which is not a contact groupoid). The theory of diierentiable groupoids (called now Lie groupoids) has been introduced by C. Ehresmann E] in 1950 in his paper on connections (where he deened the gr...

In this paper we introduce the concept of ideal elements in poe-AG-groupoid and give some characterizations and properties of their ideal elements. So we consider some results concerning ideals in poe-semigroups and investigate them in poe-AG-groupoids. Also, the class of ideal elements of poe-AG-groupoids are studied, certain intrinsic and basic properties of poe-AG-groupoids including: ideal,...