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.     

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.     

Congruence kernels of orthomodular implication algebras

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthomodular implication algebras. These are in a natural one-to-one correspondence with families of orthomodular lattices. It is proved that congruence kernels of orthomodular implication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices.     

Minimal clones—a minicourse

This paper provides an elementary introduction to minimal clones, as well as a survey of recent trends and results. Received September 27, 2002; accepted in final form October 10, 2002.     

Decomposability of free Tarski algebras

In this paper we study the direct decomposability of free Tarski algebras. We show that infinite freely generated Tarski algebras are directly indecomposable, whereas finite freely generated Tarski algebras can only be decomposed into a direct product of two factors, one of which is the two-element Tarski algebra.     

On closed sets of relational constraints and classes of functions closed under variable substitutions

Pippenger’s Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite. Received April 30, 2004; accepted in final form February 8, 2005.     

Measurement structures with archimedean ordered translation groups

The paper focuses on three problems of generalizing properties of concatenation structures (ordered structures with a monotonic operation) to ordered structures lacking any operation. (1) What is the natural generalization of the idea of Archimedeaness, of commensurability between large and small? (2) What is the natural generalization of the concept of a unit concatenation structure in which the translations (automorphisms with no fixed point) can be represented by multiplication by a constant? (3) What is the natural generalization of a ratio scale concatenation structure being distributive in a conjoint one, which has been shown to force a multiplicative representation of the latter and the product-of-powers representation of units found in physics? It is established (Theorems 5.1 and 5.2) that for homogeneous structures, the latter two questions are equivalent to it having the property that the set of all translations forms a homogeneous Archimedean ordered group. A sufficient condition for Archimedeaness of the translations is that they form a group, which is equivalent to their being 1-point unique, and the structure be Dedekind complete and order dense (Theorems 2.1 and 2.2). It is suggested that Archimedean order of the translations is, indeed, also the answer to the first question. As a lead into that conclusion, a number of results are reported in Section 3 on Archimedeaness in concatenation structures, including for positive structures sufficient conditions for several different notions of Archimedeaness to be equivalent. The results about idempotent structures are fragmentary.     

Linear subspaces,symbolic powers and Nagata type conjectures

Inspired by results of Guardo, Van Tuyl and the second author for lines in P³${\mathbb{P}}^3$, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r  -dimensional planes in Pⁿ${\mathbb{P}}^n$ for n?2r+1$n?2r+1$. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P²${\mathbb{P}}^2$ is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.     

Solving equations in groups: A survey of Frobenius' theorem

A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationx ⁿ=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.LetG be a finite group of order |G|. Leto(g) denote the order ofg( G). LetH(s, k)={xG:k|o(x)| sk} wherea/b meansa dividesb and leth(s,k)=|H(s,k)|. Using this notation the simplest of Frobenius' results states ifn/|G|, then/h(n, 1). The minimum value ofh(n, 1) is discussed in the first section. Various conditions are known to insure thath(n, 1)=n. A long standing conjecture of Frobenius states ifn=h(n, 1) thenH(n, 1) is a subgroup (where of coursen/|G|). This conjecture is valid for solvable groups, as well as for various arithmetic conditions.In the second section other divisibility conditions arising from Frobenius' Theorem are discussed. One direction covers more general arithmetic divisibility condition. Another direction has a much wider scope, involving a finite number of equations of an unspecified form and is mainly due to P. Hall. Recently some divisibility conditions involving all groups of a given order have been obtained. Divisibility conditions also hold in infinite groups, and for automorphism analogues of element order. In the next section generalizations to group characters relating back to Frobenius are given. Some of these expressions are used in analyzing properties of group representations and have applications in quantum theory. In the last section clear evidence is established for the combinatorial rather than group-theoretic nature of these results. In particular, some recent work of Snapper links the counting of solutions of equations with the cycle indices in combinatorial theory. Counting solutions of equations in the symmetric groups is also discussed.     

THE FISCHER-CLIFFORD MATRICES OF THE GROUP 26: SP 6(2)

Abstract

The smallest Fischer sporadic simple group F₂₂ has 14 maximal subgroups up to conjugacy as listed in [2] and [15]. The group 2⁶: SP ₆(2) is a maximal subgroup of F ₂₂ of index 694980. We use the technique of the Fischer-Clifford matrices to constuct the character table of the group 2⁶: SP ₆(2). The Fischer-Clifford matrices of 2⁶: SP ₆(2) are used, together with the character tables of SP ₆(2) and a maximal subgroup 2⁵:S ₆ SP ₆ (2) of index 63, to construct the character table of 2⁶: SP ₆ (2).     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

Prime spectra in modular varieties

By using the concept of modular commutator, prime congruences are defined for algebras in modular varieties. Then the prime spectrum of an algebra is defined and various spectral properties are discussed. In particular some conditions are given for the spectrum of an algebra to be homeomorphic to a ring spectrum.Presented by A. F. Pixley.     

Symmetry classes of tensors via semigroup operands

The symmetry of tensors, such as the symmetric or antisymmetric ones (built on a finite-dimensional complex vector space) may be described by a complex-valued homomorphism of the symmetric group with the specification that its action equal scalar multiplication by the value, e.g. by 1 or sign. This condition may be construed as a universalizing operand (over the symmetric group with 0) homomorphism from the unsymmetrized tensors—a restructuring which permits a clearer and more effective treatment of these symmetries; freed from the multilinear setting in which they arose, it also points the way to a development of semigroup symmetries on more general universal algebras.     

Types and coalgebraic structure

We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all F-coalgebras. In particular, we give coalgebraic characterizations for the condition that F weakly preserves pullbacks, kernel pairs or preimages. We also describe regular monos and epis. In case that |F(1)| ≠ 1 we show that F preserves preimages iff for every class of F-coalgebras. The case |F(1)| = 1 is left as an open problem.Dedicated to the memory of Ivan RivalReceived August 29, 2003; accepted in final form July 13, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.