Podsvaz

Podsvaz je podmnožina svazu, která je sama také svazem.

Definice

Nechť (A,∧,∨) je svaz a B je neprázdná podmnožina A. Pak B se nazývá podsvazem svazu A, platí-li, že B je uzavřená vzhledem ke svazovým operacím „∧“ a „∨“. tedy

Příklad

Množina B = {1,2,3,5,6,10,15,30} všech přirozených dělitelů čísla 30 je podsvazem svazu

Vlastnosti

Jestliže B je podsvaz svazu A, pak B je svazem vzhledem k indukovanému uspořádání.

Obráceně to nemusí platit. Podmnožina může být svazem, ale nemusí být podsvazem v A.

Konvexní podsvaz

Nechť (A,∧,∨) je svaz a B je podsvaz A. Pak B se nazývá konvexní podsvaz, je-li

Neboli podsvaz je konvexní, když s každými dvěma prvky obsahuje i celý interval [a,b].

Odkazy

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