Ideals of core regular double Stone algebra

2018 ◽  
Vol 11 (06) ◽  
pp. 1850083
Author(s):  
A. R. J. Srikanth ◽  
R. V. G. Ravi Kumar

A regular double Stone algebra with nonvoid core is called core regular double Stone algebra (CRDSA) [6] and this particular core element affects the behavior of the algebra in certain aspects especially in characterization of maximal and prime ideals. In this paper, an elegant characterization for maximal and prime ideals of a CRDSA is established.

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.


1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.


Blood ◽  
2013 ◽  
Vol 122 (21) ◽  
pp. 4901-4901
Author(s):  
Jean-Francois M Rual ◽  
Jay L. Hess ◽  
Tao Xu ◽  
Cailin Collins ◽  
Honglai Zhang ◽  
...  

Abstract The homeodomain-containing transcription factor HOXA9 is a core element of the HOXA9 enhanceosome, a critical DNA-protein complex that regulates hematopoietic stem cell self-renewal during hematopoiesis. Several genetic mutations observed in acute myeloid leukemia (AML) patients, including MLL translocations, are associated with aberrant up regulation of HOXA9, thus disrupting the hematopoietic balance towards leukemogenesis. While analyses of HOXA9 and cofactors have uncovered fundamental aspects of the mechanisms through which these proteins mediate their functions, questions remain. For example, what molecular mechanisms contribute to switching HOXA9 enhanceosomes off during myeloid differentiation? Could these mechanisms be targeted for the therapeutic benefit of leukemia patients? Characterization of the molecular interactions in which HOXA9 enhanceosome proteins are involved should shed light on the mechanisms that govern these proteins during both normal hematopoiesis and leukemogenesis. We recently discovered that HOXA9 interacts physically with OGT, the only O-linked N-acetyl glucosamine transferase in humans. We also demonstrated that HOXA9 is O-GlcNAcylated by OGT. Investigation of the functional relevance of this interaction to HOXA9-driven leukemogenesis is currently under way using interaction- and O-GlcNAcylation-deficient alleles of HOXA9 in a colony formation assay. Our preliminary results suggest that OGT inhibits HOXA9’s ability to transform primary bone marrow cells, thus suggesting OGT is a potential tumor suppressor of HOXA9-driven leukemogenesis. Current efforts focus on further dissecting the molecular interplay occurring between HOXA9 and OGT on chromatin, its impact on the regulation of HOXA9 targets and its role in HOXA9-driven leukemogenesis. Work is also under way to identify factors involved in the OGT-mediated regulation of HOXA9 enhanceosomes. Disclosures: No relevant conflicts of interest to declare.


1969 ◽  
Vol 21 ◽  
pp. 884-894 ◽  
Author(s):  
C. C. Chen ◽  
G. Grätzer

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).


Author(s):  
J. Catherine ◽  
B. Elavarasan

In this paper, we study the notion of $M$-ideals in partially ordered sets and examine the various properties of $M$-ideals. Further, the relations between $M$-ideals and $\alpha$-ideals in partially ordered sets are discussed. Moreover, a characterization of prime ideals to be $M$-ideals is obtained.


Author(s):  
Mohammed Issoual

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]


Author(s):  
Goulwen Fichou ◽  
Johannes Huisman ◽  
Frédéric Mangolte ◽  
Jean-Philippe Monnier

AbstractNous étudions l’anneau des fonctions rationnelles qui se prolongent par continuité surWe study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.


1981 ◽  
Vol 24 (1) ◽  
pp. 133-147 ◽  
Author(s):  
Ivo Düntsch

We prove that a regular double Stone algebra is protective in the category of Stone algebras if and only if its centre is a projective Boolean algebra and its dense set is countably generated as a filter. It follows that every countable regular double Stone algebra is projective as a Stone algebra.


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