Regular Varieties Research Articles

Pseudovarieties of completely regular semigroups

We shall show that several results concerning the lattice of completely regular semigroup varieties find their analogues for the lattice of pseudovarieties of completely regular semigroups. We establish several complete idempotent endomorphisms and a subdirect decomposition of this lattice of pseudovarieties. These investigations culminate in Theorem 18 which is the analogue for pseudovarieties of Polak’s description [19] of the lattice of completely regular semigroup varieties. We shall in particular be able to describe the lattice of pseudovarieties of orthogroups in terms of the lattice of pseudovarieties of groups.

The one-block property in varieties of semigroups

The one-block property is stated and it is proved that, whenever it holds for a class closed under homomorphic images, it implies congruence semimodularity for the whole class. An equational characterization of regular varieties having the one-block property is obtained. Some characterization is obtained also for irregular varieties of semigroups having the one-block property.

Transport Anomalies and Internal Structural Models of Stable Quasicrystals

Giant transport anomalies in (Fe,Ru)-Cu-Al quasicrystals can be explained by an internal structural model based on two large building blocks, one icosahedral and one layered. The model explains a wide variety of electronic and structural regularities.

Syntactic patterns in scientific sublanguages: A study of four disciplines

A sample of 2032 sentences from the literature of four subject disciplines was analyzed to uncover regularities and significant differences among them in their use of various syntactic features. Among those features analyzed, which are necessary parts of a well formed sentence, it was found that the two disciplines representing the social sciences rarely differ significantly from each other, and the same is true of the two hard science disciplines. However, when hard and social sciences are compared, the differences are often significant. Among syntactic features which are not necessary to a sentence, the social science disciplines still show few significant differences, but the two hard science disciplines show many more significant differences. A variety of syntactic regularities is also noted. © 1990 John Wiley & Sons, Inc.

A counterexample to the theorem of Beppo Levi in three dimensions

An embedded resolution of singularities of an algebraic variety is one which is obtained by a succession of blowing-ups with non-singular centers, so that any non-singular ambient variety remains non-singular in the process. The traditional method is to blow up non-singular subvarieties contained in the locus of points where the Hilbert-Samuel function (which for hypersurfaces is the same as multiplicity) is maximal. Then the Hilbert-Samuel function at each point of the blown-up variety is at most that of the image of that point in the original variety ([2], p. 27, Theorem (0)). The aim is to find a finite sequence of blowing-ups at the end of which the Hilbert-Samuel function decreases strictly everywhere (cf. [-1, 3, 5]). In order to achieve this, the centers of blowing-ups in such a sequence must be sufficiently large: for example, every point with maximal Hilbert-Samuel function must belong to the center at some step. Hence a natural guess is to blow up maximal non-singular subvarieties contained in the locus of points with maximal Hilbert-Samuel function. This was known to work for surfaces: let X be a hypersurface in a three-dimensional regular variety. In this case Hilbert-Samuel function and multiplicity are equivalent sets of data. Let v be the maximal multiplicity of a singular point of X and let S c X be the subvariety consisting of all the points of multiplicity v. One can show that after finitely many point blowing-ups of X we can obtain the situation when S is a normal crossings subvariety ([4, 5], pp. 519-521). Theorem of Beppo Levi ([4, 5], p. 522). Assume that S is normal crossings. Blow up any maximal non-singular subvariety of S. 7hen the v-Jold stratum remains a normal crossings subvariety and after finitely many repetitions of this procedure becomes empty. If the analogous theorem were true in higher dimensions, one could use it to obtain a simple algorithm for resolution of singularities. The point of this paper, regretfully, is to give a simple counterexample in three dimensions. I thank Boris Youssin for asking me the question and Professor H. Hironaka for an encouraging discussion.

Flasque sheaves on posets and Cohen-Macaulay unions of regular varieties

The goal of this paper is to find new examples of Cohen-Macaulay (and non-Cohen-Macaulay) rings among the coordinate rings of the unions of regular varieties. The Cohen-Macaulayness as well as other properties (seminormality, Buchsbaumness, etc.) was studied extensively for twodimensional varieties [Z, 5, 111. However, for larger dimensions not much was known besides the Reisner Theorem [lo] which gives an equivalent combinatorial condition for the coordinate ring of the unions of “coordinate” linear subspaces (Stanley-Reisner or face rings [ 123) to be Cohen-Macaulay. The method used in this paper is to represent a ring as the section ring of a sheaf on a poset and then to apply Theorem 6.4 of [ 131. More precisely, for a finite set Q of alline regular varieties we consider the poset X(a) of all intersections of elements of 0 and the sheaf &‘(a) on X(o) of the coordinate rings of these intersections. The restriction of regular functions defines a monomorphism p from the coordinate ring R of U,,, u to the section ring S of d(c). The above-mentioned result form [13] can be applied to R only if the two following restrictive conditions hold:

Residual finiteness in completely regular semigroup varieties

Residual finiteness in completely regular semigroup varieties

Mind and Chance

Much discussed but still unresolved is whether a subject's internal physical structure is a sufficient condition for his beliefs and desires. The question has sometimes been expressed as a question about microstructurally identical Doppelgänger. Imagine two subjects who are identical right down to the ions traversing the synapses. Their senses are stimulated in all the same ways, their bodies execute the same motions, and identical physical events mediate between the sensory inputs and the behavioral outputs. Must they have the very same beliefs and desires? Let us call the thesis that they must, internalism. The internalist may hold that a physical similarity less complete than this will also guarantee identity in beliefs and desires, but certainly, he holds, perfect identity of internal physical histories will suffice.Internalism will be opposed by those who sense that the nature of mentality is closely tied to the nature of explanation in terms of mental states and that in explaining a subject's behavior we cannot abstract, even in principle, from the character of the environment in which the subject is embedded. This essay offers a partial articulation of this point of view and shows how it conflicts with internalism. A consequence of the view to be described is that our attributions of belief must reflect the probabilistic regularities in the subject's environment. As we shall see, this consequence conflicts with internalism in two ways. The first conflict turns on the fact that there is no limit on the possible variety of such regularities. The second conflict turns on the fact that two subjects might by chance have the same micro-structural history though different probabilistic regularities obtain in their respective environments.

Mal'cev products of varieties of completely regular semigroups

AbstractWhilst the Mal'cev product of completely regular varieties need not again be a variety, it is shown that in many important instances a variety is in fact obtained. However, unlike the product of group varieties this product is nonassociative.Two important operators introduced by Reilly are studied in the context of Mal'cev products. These operators are shown to generate from any given variety one of the networks discovered by Pastijn and Trotter, enabling identities to be provided for the varieties in the network. In particular the join O V BG of the varieties of orthogroups and of bands of groups is determined, answering a question of Petrich.

Extensions of algebraic systems

There are several generalizations to universal algebras of the notion "The group A \mathfrak {A} is an extension of the group B \mathfrak {B} by the group C \mathfrak {C} ". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.

Extensions of Algebraic Systems

There are several generalizations to universal algebras of the notion "The group $\mathfrak {A}$ is an extension of the group $\mathfrak {B}$ by the group $\mathfrak {C}$". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.

On the Brauer group of a projective variety

This paper presents a direct, torsion-theoretic description of the Brauer group of a projective schemeX. IfX is a regular projective variety of dimension at most two, then Br(X) reduces to the relative Brauer group of the homogeneous coordinate ring ofX, based on pseudo Azumaya algebras.

Correspondence between regular varieties of unary algebras and semigroups

Correspondence between regular varieties of unary algebras and semigroups

Some criteria for the regularity of a direct product of regular p-groups

AbstractA sufficient condition, which may also be necessary, for the regularity of the direct product of two finite regular p-groups is obtained. In an attempt to decide the necessity of the condition, we look at two related questions. Firstly, does the regularity of the direct square of a regular p-group imply the regularity of all the finite direct powers; secondly, is the join of two regular varieties regular? Fainally, the results obtained are used to give some precise answers in special cases.

Regular varieties of algebras

Regular varieties of algebras

Relatively free proucts in regular varieties

The principal result states that ifV is a regular variety without operations of rank zero (distinguished elements), then any two representations of an algebraA ∈V as aV-free product of subalgebras ofA have a common refinement.

The linguistic consequences of being a lame

ABSTRACTThe most uniform and characteristic variety of Black English is the grammar used by members of the vernacular culture in pre-adolescent and adolescent years: the ‘Black English Vernacular’. Less regular varieties which show the influence of other dialects, are used by isolated individuals within the community (‘lames’) who are less familiar with vernacular norms. Rules that are categorical for the vernacular are often variable for lames. Within the vernacular peer groups, core members show the most regular or frequent use of characteristic rules such as the deletion ofis. Since most linguists grow up relatively isolated individuals within their own communities, it is important that they become aware of the linguistic consequences of being lame, and apply this knowledge in their methodology.Who's the lame who says he knows the gameAnd where did he learn to play?– The Fall

A regular variety of type <2, 2, 1, 1, 0, 0>

A regular variety of type <2, 2, 1, 1, 0, 0>

The pathology and classification of talipes.

A definition of talipes is suggested, and it is maintaned that if this definition is accepted and the different varieties of talipes analysed it can be demonstrated that the deformity is the result of moulding by mechanical forces in utero. Three regular varieties of talipes and an unlimited number of irregular ones are described.