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Neutral elements and central elements are characterized in different classes of posets such as sectionally semi-complemented
posets, atomistic posets etc.
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In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected. 相似文献
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The concept of strong elements in posets is introduced. Several properties of
strong elements in different types of posets are studied. Strong posets are characterized in
terms of forbidden structures. It is shown that many of the classical results of lattice theory
can be extended to posets. In particular, we give several characterizations of strongness
for upper semimodular (USM) posets of finite length. We characterize modular pairs in
USM posets of finite length and we investigate the interrelationships between consistence,
strongness, and the property of being balanced in USM posets of finite length. In contrast
to the situation in upper semimodular lattices, we show that these three concepts do not
coincide in USM posets. 相似文献
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The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely,
`A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for
any adjunct representation of a dismantlable lattice the number of chains as well as the number of times a pair of elements
occurs remains the same. If a dismantlable lattice L has n elements and n+k edges then it is proved that the number of irreducible elements of L lies between n-2k-2 and n-2. These results are used to enumerate the class of lattices with exactly two reducible elements, the class of lattices with
n elements and upto n+1 edges, and their subclasses of distributive lattices and modular lattices.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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The fundamental characterization theorem of standard elements in lattices is extended to posets. Several other characterizations of standard elements are obtained in a sectionally semi-complemented poset and also in an atomistic, dually sectionally semi-complemented poset. 相似文献
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For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L. 相似文献
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