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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difference between join and union
Is there a difference between the join operator, $\wedge$, and the union of a set?
In particular, what is the join of $a \wedge b $ and $b \wedge c$? Is it $a\wedge b \wedge c$ or is it $0$?
I seem ...
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How to prove that any x in a complemented distributive lattice cannot have two complements?
How can I prove the following statement?
In a complemented lattice, if there exist two complements for any x then the lattice is not distributive.
I thought of showing that, in a complemented and ...
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Lattices - How to prove a simple inequality?
Lattices are kind of new to me and I'm not yet familiar with all of their properties so excuse me if what I'm asking here is extremely basic or easy.
How can I prove the following inequality for a ...
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Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$
In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization:
A lattice is distributive if and only if ...
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Simple example of non-arithmetic ring (non-distributive ideal lattice)
Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?
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What can be said about the lattice of topologies on a given set?
Consider a set $X$ and a set $T$ of topologies on $X$. Then $(T, \leq)$ (with $\sigma \leq \tau$ if $\sigma$ is coarser than $\tau$) forms a bounded lattice with join given by intersection and meet $\...
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How to manage without specifying a particular algebraic system?
My long standing question:
How to eliminate writing $\cap^L$ instead of plain $\cap$ when we deal with more than one lattice? (and likewise with other (finite and infinite) structures) It is ...
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Basic Properties of distributive lattices
let E be a set . Let $\mathcal{D} \subseteq 2^E$ be a distributive lattice with $\phi, E \in \mathcal{D}$.
For each $ e \in E$, define
$$\mathcal{D}(e) = \cap \{ X | e \in X \in \mathcal{D}\}$$
My ...
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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"...
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topological sort of partial order into sorted sets
Given a partial order of elements, one can use topological sorting to produce a sorted list of elements. For example, if we have the partial order A->B and A->C, then the possible topological sort ...
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can GCD(0,8)≠1 be proven purely by lattice laws?
Triggered by previous question, can one prove GCD(0,8)≠1 purely by lattice laws?
Brute force Prover9/Mace4 assertions
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What is the importance of lattice based access control?
I've been reading the following papers in access control.
http://faculty.nps.edu/dedennin/publications/lattice76.pdf
http://www.winlab.rutgers.edu/~trappe/Courses/AdvSec05/access_control_lattice.pdf
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limsup and liminf of a sequence of points in a set
My ways to define/write limsup and liminf of a sequence of points in a set $X$:
They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply here....
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limsup and liminf of a sequence of subsets of a set
I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.
It says there are two different ways
to define them, but first gives what is common for the ...
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Is the Knaster-Tarski Fixed Point Theorem constructive?
According to Tarski's Fixed Point Theorem, for a complete lattice $L$, and monotone function $f:L \rightarrow L$, the set of fixed points of $f$ forms complete lattice.
Definition of $lfp(f)$ and $...
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