Another independent self-dual basis for the trivial variety

We give a new independent self-dual 3-basis for the trivial variety with two binary operations. Received October 24, 2006; accepted in final form January 25, 2007.     

A combinatorial characterization of normalizations of Boolean algebras

In this paper we prove that if a groupoid has exactly distinct n-ary term operations for n=1, 2, 3 and the same number of constant unary term operations for n=0, then it is a normalization of a nontrivial Boolean algebra. This, together with some general facts concerning normalizations of algebras, which we recall, yields a clone characterization of normalizations of nontrivial Boolean algebras: A groupoid (G;·) is clone equivalent to a normalization of a nontrivial Boolean algebra if and only if the value of the free spectrum for (G;·) is for n = 0, 1, 2, 3. In the last section the Minimal Extension Property for the sequence (2, 3) in the class of all groupoids is derived. Received September 15, 2004; accepted in final form October 4, 2005.     

Semilattices with closure

We examine the varieties of semilattices (called CSLs) with an additional closure operator treated as a unary operation. Topics investigated include the word problem in free CSLs, the lattice of CSL varieties, the finite basis problem for the identities of finite CSLs and a representation as intersection closed subsets of topological spaces.Received September 13, 2001; accepted in final form January 3, 2004.     

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.     

Hyperidentities and solid varieties

Let V be a variety of type τ. A type τ hyperidentity of V is an identity of V which also holds in an additional stronger sense: for every substitution of terms of the variety (of appropriate arity) for the operation symbols in the identity, the resulting equation holds as an identity of the variety. Such identities were first introduced by Walter Taylor in [27] in 1981. A variety is called solid if all its identities also hold as hyperidentities. For example, the semigroup variety of rectangular bands is a solid variety. For any fixed type τ, the collection of all solid varieties of type τ forms a complete lattice which is a sublattice of the lattice L(τ) of all varieties of type τ. In this paper we give an overview of the study of hyperidentities and solid varieties, particularly for varieties of semigroups, culminating in the construction of an infinite collection of solid varieties of arbitrary type. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 3, 2006. This paper is an expanded version of a talk presented at the Conference on Algebras, Lattices and Varieties in Honour of Walter Taylor, in Boulder Colorado, August 2004. The author’s research is supported by NSERC of Canada.     

Star-regularity and regular completions

In this paper we establish a new characterisation of star-regular categories, using a property of internal reflexive graphs, which is suggested by a recent result due to O. Ngaha Ngaha and the first author. We show that this property is, in a suitable sense, invariant under regular completion of a category in the sense of A. Carboni and E.M. Vitale. Restricting to pointed categories, where star-regularity becomes normality in the sense of the second author, this reveals an unusual behaviour of the exactness property of normality (i.e. the property that regular epimorphisms are normal epimorphisms) compared to other closely related exactness properties studied in categorical algebra.     

Star-linear equational theories of groupoids

We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

Few subpowers, congruence distributivity and near-unanimity terms

We prove that for any variety , the existence of an edge-term (defined in [1]) and Jónsson terms is equivalent to the existence of a near-unanimity term. We also characterize the idempotent Maltsev conditions which are defined by a system of linear absorption equations and which imply congruence distributivity. The first author was supported by the grant no. 144011G of the Ministry of Science and Environment of Serbia. The work of the second author was supported by US NSF grant no. DMS 0245622.     

On theories with the general disjunction property

A property is defined which generalizes both the disjunction property and the instantiation property of Horn clause theories. A necessary and sufficient condition is given for a theory in order to have that property and examples are used to show its applicability. Presented by M. Valeriote. Received July 12, 2005; accepted in final form January 6, 2006.     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

On the Spread of Finite Simple Groups

The spread of a nite group is the maximal integer k so that for each k non-identity elements of G there is an element generating G with each of them. We prove an asymptotic result characterizing the finite simple groups of bounded spread. We also obtain estimates for the spread of the various families of finite simple groups, and show that it is at least 2, with possibly finitely many exceptions. The proofs involve probabilistic methods.The first author acknowledges the support of the NSF; the second author acknowledges the support of the Israel Science Foundation and the hospitality of USC; both authors acknowledge the support and hospitality of MSRI.     

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.     

Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors

In this paper, we derive the Berry-Esseen bounds of the wavelet estimator for a nonparametric regression model with linear process errors generated by φ-mixing sequences. As application, by the suitable choice of some constants, the convergence rate O(n^−1/6) of uniformly asymptotic normality of the wavelet estimator is obtained. Our results generalize some known results in the literature.     

Optimal Mal’tsev conditions for congruence modular varieties

For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form August 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Finite projective planes admitting a projective linear group PSL (2, q)

Let S be a projective plane, and let G?Aut(S) and PSL(2, q) ? G ? PΓL(2, q) with q > 3. If G acts point-transitively on S, then q = 7 and S is of order 2.     

An augmenting path algorithm for linear matroid parity

Linear matroid parity generalizes matroid intersection and graph matching (and hence network flow, degree-constrained subgraphs, etc.). A polynomial algorithm was given by Lovász. This paper presents an algorithm that uses timeO(mn ³), wherem is the number of elements andn is the rank. (The time isO(mn ^2.5) using fast matrix multiplication; both bounds assume the uniform cost model). For graphic matroids the time isO(mn ²). The algorithm is based on the method of augmenting paths used in the algorithms for all subcases of the problem. First author was supported in part by the National Science Foundation under grants MCS 78-18909, MCS-8302648, and DCR-8511991. The research was done while the second author was at the University of Denver and at the University of Colorado at Boulder.     

Self-orthogonal cyclicn-quasigroups

A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x ₁, ...,x _n ),x ₁, ...,x _n–1) =x _n is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated.