The Free Distributive Semilattice Extension of a Distributive Poset

AbstractIn this paper we shall investigate the mildly distributive meet-semilattices by means of the study of their filters and Frink-ideals as well as applying the theory of annihilator. We recall some characterizations of the condition of mildly-distributivity and we give several new characterizations. We prove that the definition of strong free distributive extension, introduced by Hickman in 1984, can be simplified and we show a correspondence between (prime) Frink-ideals of a mildly distributive semilattice and (prime) ideals of its strong free distributive extension.

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Characterizations of the 0-distributive poset

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2005.10698022 ◽

2005 ◽

Vol 8 (1) ◽

pp. 81-98

Author(s):

P. Balasubramani ◽

R. Viswanathan

Keyword(s):

Distributive Poset

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Some remarks on certain classes of semilattices

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171282000039 ◽

1982 ◽

Vol 5 (1) ◽

pp. 21-30 ◽

Cited By ~ 2

Author(s):

P. V. Ramana Murty ◽

M. Krishna Murty

Keyword(s):

Necessary And Sufficient Condition ◽

Dense Element ◽

Pseudocomplemented Semilattice ◽

Distributive Semilattice ◽

Interesting Corollary ◽

Definition Of ◽

Almost All ◽

Necessary And Sufficient ◽

Distributive Semilattices ◽

Natural Way

In this paper the concept of a∗-semilattice is introduced as a generalization to distributive∗-lattice first introduced by Speed [1]. It is shown that almost all the results of Speed can be extended to a more eneral class of distributive∗-semilattices. In pseudocomplemented semilattices and distributive semilattices the set of annihilators of an element is an ideal in the sense of Grätzer [2]. But it is not so in general and thus we are led to the definition of a weakly distributive semilattice. In§2we actually obtain the interesting corollary that a modular∗-semilattice is weakly distributive if and only if its dense filter is neutral. In§3the concept of a sectionally pseudocomplemented semilattice is introduced in a natural way. It is proved that given a sectionally pseudocomplemented semilattice there is a smallest quotient of it which is a sectionally Boolean algebra. Further as a corollary to one of the theorems it is obtained that a sectionally pseudocomplemented semilattice with a dense element becomes a∗-semilattice. Finally a necessary and sufficient condition for a∗-semilattice to be a pseudocomplemented semilattice is obtained.

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Baer ideals in 0-distributive posets

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500558 ◽

2016 ◽

Vol 09 (03) ◽

pp. 1650055

Author(s):

Vinayak Joshi ◽

Nilesh Mundlik

Keyword(s):

Prime Ideal ◽

Distributive Lattices ◽

Distributive Semilattices ◽

Distributive Poset ◽

Appl Math

In this paper, we study Baer ideals in posets and obtain some characterizations of Baer ideals in [Formula: see text]-distributive posets. Further, we prove that in an ideal-distributive poset, every ideal is Baer (normal) if and only if every prime ideal is Baer (normal). We extend the concept of a quasicomplement to posets and prove characterizations of quasicomplemented poset. This extend the results regarding Baer ideals and quasicomplemented lattices mentioned in [Y. S. Pawar and S. S. Khopade, [Formula: see text]-ideals and annihilator ideals in 0-distributive lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49(1) (2010) 63–74; Y. S. Pawar and D. N. Mane, [Formula: see text]-ideals in 0-distributive semilattices and 0-distributive lattices, Indian J. Pure Appl. Math. 24(7–8) (1993) 435–443] to posets.

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Prime ideals in 0-distributive posets

Open Mathematics ◽

10.2478/s11533-013-0206-z ◽

2013 ◽

Vol 11 (5) ◽

Cited By ~ 4

Author(s):

Vinayak Joshi ◽

Nilesh Mundlik

Keyword(s):

Principal Ideal ◽

Prime Ideals ◽

Classical Theorem ◽

The Third ◽

Minimal Prime ◽

Distributive Poset ◽

Stone’S Theorem

AbstractIn the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.

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