Slim groupoids

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 groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids. The work is a part of the research project MSM0021620839 financed by MSMT.     

Some (2,n) combinatorial Stein groupoids

A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identityx(xy)=yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities.     

On Local Integrability Conditions of Jet Groupoids

A jet groupoid ℛ _q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms X→X. As a subbundle of J ^q (X,X), a jet groupoid can be considered as a system of nonlinear partial differential equations (PDE). This leads to the question if ℛ _q is formally integrable. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, which is a section ω of a natural bundle ℱ. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects. Thanks to M. Barakat and W. Plesken for discussions. The author was supported by DFG Grant Graduiertenkolleg 775.     

On some congruences of power algebras

In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).     

Lyndon’s groupoid generates a small almost Cross variety

Lyndon’s groupoid of order seven is the first published example of a non-finitely based finite algebra. The main objective of the present article is to investigate the variety generated by this groupoid and its subvarieties. It is shown that the subvarieties of form a chain of order five, all elements of which except are Cross varieties. It follows that the variety is also generated by a groupoid of order six and that any groupoid with five or fewer elements does not generate . Consequently, Lyndon’s example of a non-finitely based finite algebra could have been of order six instead of seven. It is also shown that, with respect to some important properties, Lyndon’s groupoid contrasts greatly with several well-known non-finitely based finite groupoids that were discovered shortly after its publication. Presented by R. Freese.     

Shades of orthodoxy in Rees-Sushkevich varieties

It is shown that, within the class of Rees-Sushkevich varieties that are generated by completely (0-) simple semigroups over groups of exponent dividing n, there is a hierarchy of varieties determined by the lengths of the products of idempotents that will, if they fall into a group ℋ-class, be idempotent. Moreover, the lattice of varieties generated by completely (0-) simple semigroups over groups of exponent dividing n, with the property that all products of idempotents that fall into group ℋ-classes are idempotent, is shown to be isomorphic to the direct product of the lattice of varieties of groups with exponent dividing n and the lattice of exact subvarieties of a variety generated by a certain five element completely 0-simple semigroup.     

A (2, 11) combinatorial groupoid

The variety of groupoids defined by the identitites (yx)x = xy and ((xy)(yx))(xy) = y has the properties that every groupoid generated by two elements is of order 11. The two generating identities imply others with a wide variety of combinatorial implications.     

Automorphisms and Homotopies of Groupoids and Crossed Modules

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids C\mathcal{C}, in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups C_u\mathcal{C}_u formed by restricting to a single object u. Finally, we show that the group of homotopies of C\mathcal{C} may be determined once the group of regular derivations of C_u\mathcal{C}_u is known.     

Filtering germs: Groupoids associated to inverse semigroups

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.     

Varieties of Birkhoff Systems Part I

A Birkhoff system is an algebra that has two binary operations ? and + , with each being commutative, associative, and idempotent, and together satisfying x?(x + y) = x+(x?y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bottom part of the lattice of subvarieties of Birkhoff systems, in particular the role of meet and join distributive Birkhoff systems. Our purpose in this note is to further explore the lattice of subvarieties of Birkhoff systems. A primary tool is consideration of splittings and finite bichains, Birkhoff systems whose join and meet reducts are both chains. We produce an infinite family of subvarieties of Birkhoff systems generated by finite splitting bichains, and describe the poset of these subvarieties. Consideration of these splitting varieties also allows us to considerably extend knowledge of the lower part of the lattice of subvarieties of Birkhoff systems     

The variety generated by equivalence algebras

Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x,y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base consisting of equations in three variables only. We also prove that all subdirectly irreducibles in the variety generated by equivalence algebras are embeddable into the three-element equivalence algebra, corresponding to the equivalence with two blocks on three elements. Received September 21, 1998; accepted in final form May 11, 1999.     

Varieties of Birkhoff Systems Part II

This is the second part of a two-part paper on Birkhoff systems. A Birkhoff system is an algebra that has two binary operations ? and + , with each being commutative, associative, and idempotent, and together satisfying x?(x + y) = x+(x?y). The first part of this paper described the lattice of subvarieties of Birkhoff systems. This second part continues the investigation of subvarieties of Birkhoff systems. The 4-element subdirectly irreducible Birkhoff systems are described, and the varieties they generate are placed in the lattice of subvarieties. The poset of varieties generated by finite splitting bichains is described. Finally, a structure theorem is given for one of the five covers of the variety of distributive Birkhoff systems, the only cover that previously had no structure theorem. This structure theorem is used to complete results from the first part of this paper describing the lower part of the lattice of subvarieties of Birkhoff systems.     

Distributive groupoids and biassociativity

The structure of all distributive topological groupoidsM on a closed or half-closed real interval is completely determined provided one endpoint is a zero forM, M contains an injective idempotent e, and (ex)(ye)= (ey)(xe) holds for allx, y inM. A groupoid is distributive if (xy)z = (xz)(yz) andx(yz)= (xy)(xz) hold for allx, y, z inM. An idempotente is called injective ifx =y wheneverxe =ye orex =ey. It is a corollary that all such groupoids are actually medial. The proof is accomplished by showing that, in a certain sense, distributivity corresponds to biassociativity as mediality corresponds to associativity. A groupoid is called biassociative it the subgroupoid generated by each pair of elements in the groupoid is associative.     

Discriminator or dual discriminator?

Varieties are considered with p(x, y, z), a single ternary operation, which acts as a local discriminator or dual discriminator on the subdirectly irreducible elements. If p(x, y, z) is "global", then all subvarieties are finitely based. In the general case a continuum of non-finitely based subvarieties are presented. A graph theoretical picture leads to a variety of groupoids connecting the left-zero and the right-zero semigroups. For this variety some open problems are presented. Received October 7, 1998; accepted in final form October 4, 1999.     

On the quiver-theoretical quantum Yang–Baxter equation

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.     

Left symmetric left distributive operations on a group

We study equational theories of several left symmetric left distributive operations on groups. Normal forms of terms in the variety of LSLD groupoids, LSLD medial groupoids, LSLD idempotent groupoids and LSLD medial idempotent groupoids are found. Received October 11, 2001; accepted in final form December 9, 2004.     

Groupoid cohomology and extensions

We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.

     

Commutative medial ternary groupoids

The paper studies the class of commutative medial ternary groupoids. A construction of ternary semiterms is given and it is proved that the equational theory of medial commutative ternary groupoids is solvable, namely, an algorithm is found, which in allmedial commutative ternary groupoids verifies the validity of the identity u = v for any pair (u, v) of terms. A construction of free medial commutative ternary groupoids is given, and it is proved that anymedial commutative ternary groupoid has a convex linear representation.     

Free products of unital ℓ-groups and free products of generalized MV-algebras

We generalize Komori’s characterization of the proper subvarieties of MV-algebras. Namely, within the variety of generalized MV-algebras (GMV-algebras) such that every maximal ideal is normal, we characterize the proper top varieties. In addition, we present equational bases for these top varieties. We show that there are only countably many different proper top varieties and each of them has uncountably many subvarieties. Finally, we study coproducts and we show that the amalgamation property fails for the class of n-perfect GMV-algebras, i.e., GMV-algebras that can be split into n + 1 comparable slices. This paper has been supported by the Center of Excellence SAS -Physics of Information-I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contracts No. APVT-51-032002, APVV-0071-06, Bratislava.