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1.
Shelly L. Wismath 《Algebra Universalis》2006,55(2-3):305-318
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. 相似文献
2.
K. Auinger 《Algebra Universalis》1995,34(2):185-190
It is shown that an existence varietyV of regular semigroups contains (all) free products if and only ifV consists solely of locally inverse orE-solid semigroups.Presented by B. M. Schein.The author is indebted to the Australian Research Council for financial support (ARC grant A69231516). 相似文献
3.
It is proved that the operator Lie algebra ε(T,T∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q∗)<+∞, where ε(T,T∗) denotes the smallest Lie algebra containing T,T∗, and A(Q,Q∗) denotes the associative subalgebra of B(H) generated by Q,Q∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator. 相似文献
4.
Clifford Bergman 《Order》1989,6(1):49-58
We prove that if v is the variety generated by a finite modular lattice, then v is not an elementary class. We also consider the same question for the variety generated by N
5.Research partially supported by National Science Foundation grant DMS-8701643. 相似文献
5.
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program. 相似文献
6.
Denote by RS
n
the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS
n
are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups
are said to be exact. For each positive integer m, define C
m
RS
n
to be the class of all semigroups S in RS
n
with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup.
The classes C
m
RS
n
constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties
of C
2
RS
n
is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties
over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core
with any subgroup, if nonempty, is contained in the center of that subgroup.
The equational property of the varieties C
m
RS
n
is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C
m
RS
n
, where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C
∞
RS
n
) is non-finitely based. Further, the variety C
∞
RS
n
contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately
finitely based and non-finitely based.
Received October 29, 2003; accepted in final form February 11, 2007. 相似文献
7.
We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A2 and B2=C2 and prove that no such collection exists for type G2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2. 相似文献
8.
We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras. 相似文献
9.
Toric varieties,lattice points and Dedekind sums 总被引:8,自引:0,他引:8
James E. Pommersheim 《Mathematische Annalen》1993,295(1):1-24
10.
We compare the two recently introduced semidirect product operations
*r and
*rr within the lattice of e-varieties of locally inverse
semigroups. For each e-variety
which
contains all rectangular bands and is properly contained in the e-variety of all completely
simple semigroups, the inclusions
are proved where
is the e-variety of all semilattices and
the variety of all abelian
groups of exponent dividing q where q is any
integer greater than one. Some consequences for the class of finite locally inverse semigroups are also obtained. 相似文献
11.
Raymond Broeksteeg 《Semigroup Forum》1994,49(1):335-348
We define a bivariety of regular biordered sets to be a nonempty class of regular biordered sets which is closed under taking
direct products, regular bimorphic images and relatively regular biordered subsets. It is then shown that there is a complete
lattice morphism mapping the complete lattice of all e-varieties of regular semigroups onto the complete lattice of all bivarieties
of regular biordered sets; as a corollary we prove that there is a complete lattice morphism mapping the complete lattice
of all e-varieties of E-solid regular semigroups onto the complete lattice of all varieties of solid binary algebras. Examples
of bivarieties include the class of all solid regular biordered sets and the class of all local semilattices. For each setX with at least two elements, we show that a bivariety contains a free object onX if and only if it consists entirely of solid regular biordered sets or it consists entirely of local semilattices.
The author gratefully acknowledges the financial support of an Australian Postgraduate Research Award. 相似文献
12.
Charles Buehrle 《Journal of Pure and Applied Algebra》2010,214(5):689-700
We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1,…,xn,n] to give a new construction of the Kazhdan-Lusztig representations of Sn. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x1,1,…,xn,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations. 相似文献
13.
Pierre-Emmanuel Chaput 《Indagationes Mathematicae》2003,14(2):169-182
In this paper, I give two very direct proves of the correspondance between a geometric object (Scorza varieties) and an algebraic one (Jordan algebras). I also give a short proof of the homogeneity of Scorza varieties, and a new and very simple proof of properties of the automorphism group of a Jordan algebra. 相似文献
14.
Libor Barto 《Journal of Pure and Applied Algebra》2007,211(3):721-731
A category is said to be alg-universal if every category of universal algebras can be fully embedded into it. We prove here that the category of varieties and interpretations, or in other words, the category of abstract clones and clone homomorphisms, is alg-universal. 相似文献
15.
Pierre-Emmanuel Chaput 《Mathematische Zeitschrift》2002,240(2):451-459
R. Hartshorne conjectured and F. Zak proved (cf [6,p.9]) that any smooth non-degenerate complex algebraic variety with satisfies denotes the secant variety of X; when X is smooth it is simply the union of all the secant and tangent lines to X). In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions
and . I want to give a different proof of a theorem of F. Zak classifying all Severi varieties. F. Zak proves that there exists
only four Severi varieties and then realises a posteriori that all of them are homogeneous; here I will work in another direction:
I prove a priori that any Severi variety is homogeneous and then deduce more quickly their classification, satisfying R. Lazarsfeld
et A. Van de Ven's wish [6, p.18]. By the way, I give a very brief proof of the fact that the derivatives of the equation
of Sec(X), which is a cubic hypersurface, determine a birational morphism of . I wish to thank Laurent Manivel for helping me in writing this article.
Received in final form: 29 March 2001 / Published online: 1 February 2002 相似文献
16.
Varieties of idempotent semirings with commutative addition 总被引:3,自引:0,他引:3
The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13
distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the
variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings
free in these varieties are given.
Received April 22, 2004; accepted in final form June 3, 2005. 相似文献
17.
William A. Lampe 《Periodica Mathematica Hungarica》1996,32(1-2):65-75
We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday 相似文献
18.
In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in and computed their index using certain graphs, which we call type-A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, have been defined by Panyushev (2001) [9] for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types B and C has been developed by the authors. In this article, we consider the most difficult and interesting case of type . Some new phenomena occurring here are related to the fact that the Dynkin diagram has a branching node. 相似文献
19.
John Pedersen 《Algebra Universalis》1994,31(1):95-103
Certain varieties similar to commutative semigroups are shown to have uniformly solvable word problem for all finite presentations by a confluence-completion method.Presented by H. P. Gumm. 相似文献
20.
The lattice of all subvarieties of the variety generated by all ordered bands is obtained. This lattice is distributive and
contains 78 varieties precisely. Each of these is finitely based and generated by a finite number of finite ordered bands. 相似文献