scholarly journals Characterizations of the $0$-distributive semilattice

2003 ◽  
Vol 128 (3) ◽  
pp. 237-252
Author(s):  
P. Balasubramani
Keyword(s):  
Distributive Semilattice

Notes on mildly distributive semilattices

2017 ◽  
Vol 67 (5) ◽  
Author(s):  
Sergio Arturo Celani ◽  
Luciano Javier González
Keyword(s):  
Prime Ideals ◽  
Distributive Semilattice ◽  
Definition Of ◽  
Distributive Semilattices ◽  
Distributive Extension

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.

scholarly journals Some remarks on certain classes of semilattices

1982 ◽  
Vol 5 (1) ◽  
pp. 21-30 ◽  
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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