Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

We provide a partial result on Taylor’s modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analog for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence n-permutability for a fixed n, and satisfying a non-trivial congruence identity.     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Quasiorder lattices of varieties

The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.     

Varieties of Posets

We introduce and investigate the notion of a homomorphism, of a congruence relation, of a substructure of a poset and consequently the notion of a variety of posets. These notions are consistent with those used in lattice theory and multilattice theory. There are given some properties of the lattice of all varieties of posets.     

Zero-one laws in finite W1-algebras

The authors study the notion of independence in the noncommutative analogue of a probability space, namely, a finite W¹-algebra with trace τ. Two definitions of independence, both generalizing the classical definition, are considered, and examples are given to illustrate that the definitions are inequivalent. For one of these notions of independence, the structure of a finite W¹-algebra generated by an independent family of subalgebras is determined. As a consequence, the authors obtain extensions of the Kolmogorov and Hewitt-Savage zero-one laws.     

A class of formal languages

We deal with languages that are classes of fully invariant congruences on free semigroups of finite rank. The question is posed as to whether a given fully invariant congruence coincides with a syntactic congruence of the language in question. If all classes of a given fully invariant congruence are rational languages, the corresponding variety is then said to be rational. A number of properties of rational varieties is established—in particular, we point to the way in which they are related to finite varieties. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 478–492, July–August, 1998.     

A Note on the Distributivity of the Huq Commutator Over Finite Joins

We show that the Huq commutator distributes over finite joins, in any semi-abelian algebraically cartesian closed category. As a consequence we show that for semi-abelian varieties of universal algebras (more generally for semi-abelian categories with large directed colimits of subobjects preserved, for each object B, by the functor B×??), the distributivity of the Huq commutator over joins is equivalent to algebraic cartesian closedness.     

Geometric stability theory for μ-structures

We introduce a notion of μ-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures (introduced by Krupinski in [9]). Using the Haar measure of locally compact groups, we introduce an independence, called μ-independence, in μ-structures having good properties. With this independence notion, we develop geometric stability theory for μ-structures. Then we see some structural theorems for compact groups which are μ-structure. We also give examples of profinite structures where μ-independence is different from nm-independence introduced by Krupinski for Polish structures.In an appendix, Cohen and Wesolek show that a profinite branch group gives a small action on the boundary of a rooted tree so that this actions provides a small profinite structure on the boundary of a rooted tree.     

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.     

A transference method in quantum probability

Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L₁, where some recent results by the first-named author can be used. We find applications on noncommutative Khintchine/Rosenthal type inequalities and on noncommutative Lp embedding theory.     

Permutation modules of profinite groups

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.     

Steenrod operations on the Chow ring of a classifying space

We use the Steenrod algebra to study the Chow ring CH^*BG of the classifying space of an algebraic group G. We describe a localization property which relates a given G to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and their fusion.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

Amenability of semigroups and their algebras modulo a group congruence

We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups.     

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.     

Periods in strings

In this paper we explore the notion of periods of a string. A period can be thought of as a shift that causes the string to match over itself. We obtain two sets of necessary and sufficient conditions for a set of integers to be the set of periods of some string (what we call the correlation of the string). We show that the number of distinct correlations of length n is independent of the alphabet size and is of order n^logn. By using generating function methods we enumerate the strings having a given correlation, and investigate certain related questions.     

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.