We give a simple example of a variety of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences.
Using probabilistic methods of Erdos, we construct an infinite sequence of finite graphs with arbitrarily large chromatic number, such that each is a bounded morphic image of and has no odd cycles of length at most . The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the satisfies arbitrarily many axioms from a certain axiomatisation of , while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.
Varieties are considered with p(x, y, z), a single ternary operation, which acts as a local discriminator or dual discriminator on the subdirectly irreducible elements.
If p(x, y, z) is "global", then all subvarieties are finitely based. In the general case a continuum of non-finitely based subvarieties
are presented. A graph theoretical picture leads to a variety of groupoids connecting the left-zero and the right-zero semigroups.
For this variety some open problems are presented.
Received October 7, 1998; accepted in final form October 4, 1999. 相似文献
In this paper we initiate the study of composition operators on the noncommutative Hardy space , which is the Hilbert space of all free holomorphic functions of the form
We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose to be the Grassmannian parameterizing linear subspaces of dimension d in , where k is an algebraically closed field of characteristic . Let E be a uniform vector bundle over G of rank . We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle or its dual by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in . Furthermore, we generalize the Grauert–Mlich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable bundle over the complete flag variety splits as a direct sum of special line bundles. 相似文献
By Aguglia et al., new quasi-Hermitian varieties in depending on a pair of parameters from the underlying field have been constructed. In the present paper we study the structure of the lines contained in and consequently determine the projective equivalence classes of such varieties for odd and . As a byproduct, we also prove that the collinearity graph of is connected with diameter 3 for . 相似文献