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What is an interval of a lattice in order theory?

In the Wikipedia page of modular lattices, there is a theorem that uses the notion of interval (diamond isomorphism), but the term interval is not defined there and the definition is hard to find on Google because the search results are mostly about lattices of intervals.

Can someone provide the definition?

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    $\begingroup$ Interval $\endgroup$
    – plop
    Nov 24, 2021 at 12:39

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The answer given (and already accepted) gives what seems to be the definition of interval of a poset, according to its Wikipedia article which is also linked to in a comment.

I was amazed that people call intervals to such subsets of posets;
as far as I am concern, a subset $C$ of a poset $P$ satisfying the property $p \in C$ whenever $a,b \in C$, $p \in P$ and $a\leq p \leq b$ is called a convex set.

This is the definition given in Introduction to Lattices and Order, by Davey and Priestley, page 63 (in the second edition) and also in General Lattice Theory, by Grätzer, page 21 (in the second edition too).

These references seem never to define formally what is an interval, but always refer to an interval as a closed interval, $[a,b]$, an open interval, $(a,b)$ or an half-open interval, $(a,b]$ or $[a,b)$, where $a\leq b$. (I think that Davey and Priestley only refer to closed intervals, but I'm not sure.) Here, $$(a,b) = \{x : a < x < b\},$$ $$(a,b] = \{x : a < x \leq b\},$$ $$[a,b) = \{x : a \leq x < b\},$$ while the closed interval is, of course, defined as in the other answer.

Apparently the definition is not standard.
See also the ProofWiki definitions of Convex set and Interval (coincide with the ones I referred to).

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The definition of interval for partial orders is a subset $S\subseteq X$ such that, for all $x,y\in S$ and for all $z\in X$ such that $y\le z\le x$, $z\in S$.

There is a general notion of closed/open/half-open interval in, say, $[a,b]=\{x\in X\,:\, a\le x\le b\}$. Notice, however, that these are not necessarily intervals in the sense I've mentioned. The wikipedia page is referring to this kind of "interval".

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  • $\begingroup$ For completeness, could you please add the definition of $[a,b]$? $\endgroup$ Nov 24, 2021 at 12:58

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