Introduction
Abstract
En
A characterization of v-irreducible elements and of strongly v-irriducible elements of a distributive lattice
was given by D.Drake and W.J.Thorn in (1).Among other things in (1)it was proven that an element
is v-irreducible if one can identify
, by means of a lattice isomorphism f, with a sublattice
of the power set
of a suitable set X, in such a way that
is the closure in
of an element
is the minimum element in
,with respect to the set inclusion, including x).
As is well-known an element of a distributive lattice is v-irreducible iff it is v-prime.This property is exploited is an essential manner in (1).Now then in our paper we took this property as a starting point for a characterization cf v-prime and of strongly v-prime elements of any partially ordered set (in particular of any lattice).Here,on the analogy of some characterization of v-prime elements and of strongly v-prime elements of a lattice, an element c of a partially ordered set(shortly "poset"
) is said v-prime if the subset
is v-directed, i.e.
or for every
(for every(Error rendering LaTeX formula))there exists
such that
for every
); moreover c is said strongly v-prime if
or
has maxium element.Then we prove than an element
is v-prime in
if we can identify
by means of an order isomorphism f,with a set (but not necessarily a lattice) of sets of the type of (1) in such a way that
is the closure in
of a element of
;moreover we prove that c is strongly v-prime in
if for all function f of the above type the set
is the closure in
of an element of
.
It
D.Drake e W.J.Thorn hanno in (1)una caratterizzazione degli elementi v-irriducibili e degli elementi fortemente v-irriducubili di un reticolo distributivo
.tra l'altro in (1)è stato provato che un elemento
è irriducibile se e solo se si può identificare
,tramite un isomorfismo reticolare f,con un sottoreticolo
del reticolo delle parti
di un opportuno insieme X in tal modo che
è la chiusura di
di un certo elemento
(cioè
è il più piccolo elemento di
,rispetto all'inclusione insiemistica, cui appartiene x).Come è ben noto un elemento di un reticolo distributivo è v-irriducibile se e solo se esso è v-primo.Questa propietà è usata in maniera essenziale in (1).In questo lavoro noi prendiamo lo spunto da questa propietà per dare una caratterizzazione degli elementi v-primi e degli elementi fortemente v-primi di un qualsiasi insieme parzialmente ordinato(in particolare di qualsiasi reticolo
). Precisiamo che quì, in analogia con una caratterizzazione degli elementi v-primi e degli elementi fortemente v-primi di un reticolo,un elemto c di un insieme parzialmente ordinato (
è detto v-primo se il sottoinsieme
è v-diretto, cioè
oppure per ogni
per ogni(Error rendering LaTeX formula))esiste
tale che
e
(
per ogni
);inoltre c è detto fortemente v-primo se
oppure
è dotato di massimo.Allora noi proviamo che un elemento
è v-primo in
se e solo se possiamo identificare
, tramite un isomorfismo f rispetto all'ordinamento, con un insieme di insiemi(non necessariamente un reticolo di insiemi) del tipo (1) in modo tale che
è la chiusura in (
di un elemento di
);inoltre proviamo che c è fortemente v-primo in
se e solo se per ogni isomorfismo f del tipo su menzionato l'insieme
è la chiusura in
di un punto di
.
A characterization of v-irreducible elements and of strongly v-irriducible elements of a distributive lattice
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It
D.Drake e W.J.Thorn hanno in (1)una caratterizzazione degli elementi v-irriducibili e degli elementi fortemente v-irriducubili di un reticolo distributivo
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DOI Code:
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