# Questions tagged [lattice-orders]

Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.

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### Sublattice VS lattice under the ordering of its superset

Yet another question coming from the book Lattice Theory : Foundation, written by George Grätzer. In the third set of exercises, he proposes the following one : Find a subset $H$ of a lattice $L$ ...
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### How to determine if given lattice is distributive or not?

I am trying to understand how to determine whether given lattice is distributive. I came across following: A lattice is distributive if and only if none of its sublattices is isomorphic to $M_3$ or ...
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### How may one compute the proportion of isomorphisms among the total quantity of different combinations of element sequences and associative groupings?

If I have n elements in each possible sequence with each possible Tamari lattice grouping of those sequences with respect to a non-associative commutative operation, how may one compute the proportion ...
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### Equivalent(?) Order-Theoretic Definition of a Lattice

Lattice Definition: A (partially) ordered set $(A,\preceq)$ is defined to be a lattice when every two-element subset of $A$ has both a meet and a join (i.e., a greatest lower bound and a least upper ...
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### Ideal generated by a subset of a lattice

In George Grätzer book, Lattice Theory : Foundation, there is some fundamental result I am not sure to interpret correctly. The owner of the book will find it page 32. It is the second statement of ...
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### What is the preferred convention for denoting Tamari lattice groupings?

Is there a common method or standard for denoting Tamari lattice/associative groupings with a character length less than that of the sum of the quantity of elements and parenthesis to be denoted? I ...
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### Draw a tree on top of a semi-lattice

This question is rather vague, but I hope you might have some input. I'd like to create the following structure (draw it): Imagine a semi-lattice drawn on a plane in 3d space. Now I would like to ...
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### Are all lattices chains?

Are all lattices chains? I think that is true because a chain is a poset where we can compare any two elements. A lattice is a poset where every subset has a lub and a gld. So, by reducing the size of ...
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### Finite lattice dimension and planarity

I have been struggling for several days on this question, now it is time for you folks to enlighten me :) In the book Lattice Theory: Foundations from George Gratzer (pdf file), it is stated, page 9, ...
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### Number of join-irreducibles and rank in a finite lattice

Let $(L, r)$ be a finite, ranked lattice. Is $$r(x) = \#\{j \leq x : j \in L \text{ is join-irreducible}\}$$ for all $x \in L$? This may be a naive question, though my lattice theory is weak.
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### Whitman's condition for lattice polynomials

Recall that Whitman's condition for a lattice $L$ (with join $\sqcup$ and meet $\sqcap$) says that given $a, b, c, d \in L$, it holds that $a \sqcap b \sqsubseteq c \sqcup d$ iff at least one of the ...
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### How can I find the Bipartite Order associated to a Lattice?

I am interested in representing a lattice as a tree. The only paper I have found about that is Bit-vector encoding for partially ordered sets. The author does not explain thoroughly what I want. I ...
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### Ordered Set (Steven Roman)

How do I show 1. and 2.? Let $P$ be a poset. Prove that the down map is an order embedding from $P$ into $\wp (P)$. Let $I_n = \{ 1,...,n \}$ and let $f: \wp (I_n) \rightarrow \{ 0,1 \} ^n$ be ...