Mathematics Stack Exchange
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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.

Lattices are partially ordered sets such that for any two elements $x$, $y$ there is a supremum $x\vee y$ and infimum $x\wedge y$ of the set $\{x,y\}$. Lattice theory is an important subfield of order theory.

See the Wikipedia entry for more information.

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Difference between lattice and complete lattice

Definition of lattice require that any two elements of lattice should have LUB and GLB, while complete lattice extends it to, every subset should have LUB and GLB. But by induction , it is possible to show that if any two elements have LUB and GLB…
chinu
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Example of a bounded lattice that is NOT complete

I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete? Thank you
user116732
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Find a lattice with exactly three congruence relations

A lattice $L$ is a partially ordered set such that any two elements $a$ and $b$ have a least upper bound $a\lor b$ and a greatest lower bound $a\land b$. A congruence relation on $L$ is an equivalence relation $\sim$ on $L$ that is compatible with…
PatrickR
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Complete lattices on $\mathbb{Q}$ and $\mathbb{R}$ ordered by $\leq$

To quote from my lecture notes: When every subset of A has a lub and glb, we say that the order is a complete lattice, but this takes us beyond the syllabus. It is notable that $\mathbb{Q}$, ordered by $\leq$, is not a complete lattice but…
8128
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Complete lattices and sublattices – which requirement is more stringent?

I'm studying from Michael Carter's "Foundations", and on page 29 he makes the comment, "Note that the requirement of being a sublattice is more stringent than being a complete lattice in its own right". This seems counterintuitive. Some lattice $L$…
jefflovejapan
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Proving that double negation elimination implies law of excluded middle, in Heyting algebra

Let a Heyting algebra $H$ be given. Suppose $x \in H$. Prove the following: if $\neg \neg x \cong x$, then $x \vee \neg x \cong \top$. (To be precise: here $\top$ means the (essentially unique) top element of the lattice, and negation $\neg$ is…
Dorry
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A Lattice that is not a Complete Lattice

I was studying about Complete Lattices from the text book J.P. Tremblay. The definition of a complete lattice is that each of its non empty subsets should have a least upper bound and greatest lower bound. By this definition every finite lattice…
Deepu
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Existence and uniqueness of particular binary operations on a lattice

Let $(A, \le)$ be a lattice. Consider the following properties for a commutative operation $\cdot$ on $A$: $$c \cdot (a \wedge b) = (c \cdot a) \wedge (c \cdot b)$$ $$c \cdot (a \vee b) = (c \cdot a) \vee (c \cdot b)$$ $$a \cdot b = (a \wedge b)…
Luca Bressan
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Definition of Sub-Lattice

In reference to Sub-lattices and lattices. In Wikipedia, it's given that- A sublattice of a lattice L is a nonempty subset of L that is a lattice with the same meet and join operations as L. That is if L is a lattice and $M ≠ {\displaystyle…
user9014873
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Complete Lattice and fixed point

I am wondering how to show: An order-preserving map $f$ of a complete lattice $A$ into itself has at least one fixed element.
Allitee
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Are the real numbers a lattice?

A lattice is a set with a partial order, where every pair has a unique upper and lower bound. As far as I can tell, there is nothing in the definition that forces the set to be discrete. In particular, the real numbers with their usual partial order…
prdnr
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Sub-lattices and lattices.

I have read in a textbook that $ \mathcal{P}(X) $, the power-set of $ X $ under the relation ‘contained in’ is a lattice. They also said that $ S := \{ \varnothing,\{ 1,2 \},\{ 2,3 \},\{ 1,2,3 \} \} $ is a lattice but not a sub-lattice. Why is it…
Jayasree Thomas
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Is the codomain of a surjective lattice-homomorphism Heyting if the domain is Heyting?

Let $L$ and $L'$ be lattices and let $F:L\to L'$ be a surjective lattice-homomorphism in the sense that $F$ respects $\wedge$ and $\vee$. I managed to prove for several properties of $L$ that they will be inherited by $L'$. Let me mention some: if…
drhab
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Proving that idempotence follows from other lattice axioms

I am supposed to prove that $x\wedge x = x$ and $x\vee x = x$ follow from the other lattice axioms (associativity, commutativity and absorption). So far I have $$x = x\vee(x\wedge x) = x\vee(x\wedge(x\wedge(x\vee x))) = x\vee ((x\wedge…
Cubic
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An order isomorphism between lattices is a lattice isomorphism

Suppose that $(X, \leq_X)$ and $(Y, \leq_Y)$ are ordered sets. Let $T:(X, \leq_X) \rightarrow (Y, \leq_Y)$ be an order isomorphism. Is it true that $T$ is a lattice isomorphism? I have this question because I couldn't follow the explanation here by…
Idonknow
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