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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Let $L$ be a lattice. Is there a union closed family of subsets of $[n]$ ordered by inclusion that correspond to $L$.

Let $\mathcal{C}$ be a collection of subsets of $[n]$ that is closed under taking unions (including taking the union of the empty set which is the empty set). Then $(\mathcal{C},\subseteq)$ is a ...
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Understanding Tarski's fixed-point theorem.

I changed my question slightly. (Tarski Fixed Point Theorem). Let $X=\prod^{N}_{i=1} X_{i}$ where each $X_{i}$ is a compact interval of $\mathbb{R}$. Suppose $\phi : X \rightarrow X$ is an increasing ...
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What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound?

What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound? This is an on-a-review sheet for my final. I thought the completeness axiom ...
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In a compact metric space, every pair of homeomorphic open subsets has isomorphic basis?

Let X be a compact metric space, V and W be open subsets. Suppose there is an homemorphism from V to W. Let B be a countable basis for X and B(V), B(W) the relativized basis for V and W, respectively. ...
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Are de Morgan algebras or lattices always bounded?

In some textbooks, de Morgan lattices are defined to be bounded distributive lattice satisfying the involution law and the de Morgan's laws. But in some textbooks, there is no requirement for ...
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Join in Lattice of Subobjects

In an elementary topos the join $A \vee B$ of two subjects $A \to X$ and $B \to X$ is defined to be the image of the induced morphism $A \sqcup B \to X$. For sets it holds, that this is the same as ...
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Example of non distributive module [closed]

I'm looking for a module that has three submodules $P,Q,R$ such that $P\cap (Q+R) \neq P\cap Q +P\cap R $ I'm struggling to find this example because I'm not that familiarized in module theory Can ...
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How is it the case that: Any complete lattice is a Boolean algebra.

In the book “A Functorial Model Theory” by Nourani (pg152), it is stated that However, I didn’t understand what does he mean? Because a complete lattice is not even necessarily distributive whereas ...
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Duality in Lattices

I am studying Lattices using the algebraic definition i.e. A set with 2 binary operations $\wedge , \vee$ that satisfies: Commutative of $\wedge$ and $\vee$ Associativity of $\wedge $ and $\vee$ ...
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Why is product of infimum and supremum of a commuting pair of elements in a lattice-ordered group equal to the product of the elements?

Context: Self-study. Seth Warner's Modern Algebra (1965), question $15.11$ gives: If $(G, \circ \preccurlyeq)$ is a lattice-ordered group and if $x$ and $y$ are commuting elements of $G$, then $$\sup ...
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Some question about lattice rank.

I found this equation while looking at the "Rank of a partially ordered set" "A lattice with a rank function ρ is (upper) semi-modular if:ρ(x)+ρ(y)≥ρ(x∨y)+ρ(x∧y)" (https://...
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element in longest chain and longest antichain

In an arbitrary finite poset, is there necessarily an element that is both in the longest chain and in the largest antichain? I think it is true. Longest chain contains an element of every largest ...
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Reading off semi-lattice diagram

I'm reading a chapter about mereology (in a handbook of linguistics), and I have some questions. A prepring is available here (p. 519). The symbol $\leq$ is to be interpreted as non-strict parthood, ...
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Semilattice whose Subsets are All Closed -- does it have a special name?

Context: self-education. I am currently getting my head round semilattices. My understanding is that a semilattice $(S, \odot)$ is a semigroup whose operation $\odot$ is both commutative and ...
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How do you prove that a formally constructed total ordering is unique?

Seth Warner's Modern Algebra Exercise $14.22$ gets us exploring the properties of semilattices, in particular join semilattices. Context: self-education. The specific question that has been given is: ...
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Is there an intrinsic lattice-theoretic generalization of the set of principal ideals of a ring?

Let $R, S$ be unital commutative rings. Upon revisiting basics of ring theory, I've been wondering whether there is an intrinsic lattice-theoretic description of what it means to be a principal ideal. ...
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Applications of bounded cocomplete semilattices

Call an ordered set a poset when the ordering is transitive and antisymmetric. Call a poset bounded when it has a top and a bottom element (i.e. a greatest and least element). Call a poset cocomplete ...
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Is there a general poset representation that specializes to power set lattices in case of finite boolean algebras?

I read here that every finite, complemented, distributive lattice is isomorphic to a power set lattice. Is there a general order preserving mapping from a poset $P$ to a set inclusion poset $S$, such ...
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Is a Hetying algebra with coexponetials Boolean?

Suppose we have a Heyting algebra $\mathcal A$ with coexponentials. Specifically, for every $a, b : \mathcal A$ we have an object $b \backslash a$ with the properties that $b \le a \lor (b \backslash ...
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Unique Complemented lattice which is not distributive lattice!

I know that : "if a distributive lattice is also complemented lattice then lattice should only have unique complement" but i am not able to find a case where A lattice is uniquely ...
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Help calculating relative pseudo-complements in a Heyting algebra/lattice

I'm trying to work some examples of relative pseudo-complements in lattices, to make sure I understand them. I wonder if anybody could check my examples, and tell me if I'm correct or if I've ...
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Bases and weights of Lawson topology

Theorem: For any domain $L$ one has $w(L)=w(\Sigma L)=w(\Lambda L)=w($Id$ B)$ for any basis $B$ of $L$ with $card(B)=w(L)$. In proof of $w(\Lambda L)\le w(L)$:Let $B$ be a basis of $L$ with $B=w(L)$. ...
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Can we have complete lattice "inside of" something that is not a lattice?

Can we have complete lattice "inside of" something that is not a lattice? Say we have a partially ordered set $P$. Can we have a complete lattice $Q$, which has ordering inherited from $P$, ...
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Closure of a fuzzy relation

In paper Special properties, closures and interiors of crips and fuzzy relations https://doi.org/10.1016/0165-0114(88)90126-1, can anyone explain what is the point of introducing the closure of a ...
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How is the dual notion of an extensive map called?

While reading about closure operators I encountered the following notion: Definition. Let $A$ be a partially ordered set. A map $f$ from $A$ to $A$ is extensive if $a ≤ f(a)$ for every element $a$ of ...
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Is there a polyhedron whose face lattice is a given lattice with the "diamond property"?

I learned that every $(k-2)$-face is contained in exactly two facets in a $k$-dimension polyhedra from 'Theory of linear and integer programming'. So every face lattice of polyhedra satisfies the &...
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Why inclusion-exclusion principle fails for vector subspaces

Let $U,V,W$ be three vector subspaces of a same vector space. It is well-known that $$\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$$ works but $$ \dim(U +V + W) = \dim U + \dim V + \dim W - \dim (...
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Special lattices

In a lattice one has a meet and a join operation. I would like to know more about lattices that have the following additional property: $b\leq (a \vee a')$ is equivalent to $(b\leq a \ or\ b\leq a')$...
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A general setting to state the isomorphism theorems for (complete) (semi)lattices

Lattices, complete lattices and (complete) lower and upper semilattices are all very similar algebraic structures. The homomorphisms between lattices of the same type and also the Isomorphism Theorems ...
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Left adjoint to the inclusion of Boolean algebras into distributive lattices

Let $\mathbf{Boole}$ be the category of Boolean algebras. Let $\mathbf{BDL}$ be the category of bounded distributive lattices. There is a fully faithful functor ${\mathbf{Boole} \rightarrow \mathbf{...
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Example of a lattice which is distributive but not bounded

Distributive lattices may or may not be bounded. Can someone give an example of a distributive lattice which is not bounded?
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Semi-lattices whose Hasse Diagrams are trees after transitive reduction?

Is there a name or anything else known for a semi-lattice whose Hasse Diagram becomes a tree after applying transitive reduction? Trying to find more about it since it comes up in an optimization ...
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Show that a distributive lattice can be embedded into product of two-element lattices

I have seen this exercise in Bergman: Universal Algebra: Fundamentals and Selected Topics. Let $L$ be a distributive lattice and let $2$ be a 2-element lattice. Show that there is a set $J$ and ...
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complete semilattice and its not subsemilattice/a counter-example [closed]

Is there an example of a complete semi-lattice and its subset which is itself a complete semi-lattice without being sub-semi-lattice of the bigger one ?
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Equivalence between two definitions of Lattice (order theory)

I have come across two different definitions of a lattice. The first one is that a lattice is a partially ordered set with every pair of elements having an infimum and a supremum. The other definition ...
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On Isomorphic Lattices

I've been recently studying lattices and other ordered structures (after a long break in university) and I'm stuck at a question... tried hard on this one, but all I got was a huge headache. Question: ...
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Is there a name for graphs that appear as Hasse diagrams of finite lattices?

Hasse diagrams are mathematical diagrams used to represent finite partially ordered sets, and may be seen as a kind of graph. Apparently, there are some relations between particular kinds of lattices ...
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What is a relation of category of complete sublattices to category of lattices?

What is a relation of category of complete sublattices to category of lattices? I haven´t found much about a category of lattices, but I assume objects = lattices, morphisms = lattice homomorphisms. ...
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If $(f, g)$ is a Galois connection between two bounded lattices, then if $T$ is an ideal we have $f^{-1}(T)$ is an ideal

Let $\mathcal{I}(L)\:$ and $\mathcal{I}(N)\:$ be the ideal lattices of the bounded lattices $L$ and $N$ and let $(f, g)$ be a Galois connection between $L$ and $N$, then show that $\:\forall \:\: T \...
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Is a finite lattice where each element has exactly one complement distributive? Why or why not?

While reading the paper LATTICES WITH UNIQUE COMPLEMENTS by R. P. DILWORTH, I get to know that any number of weak additional restrictions are sufficient for a lattice with unique complement to be a ...
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2 answers
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Defining arbitrary join on the set of complete ideals of a Heyting Algebra

Given a Heyting Algebra $H$ we define a complete ideal (or c-ideal) $I$ to be a subset of $H$ satisfying. $\bot \in I$ $b \in I$ and $a \leq b$ implies $a \in I$ $X \subseteq I$ and $\bigvee X$ ...
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let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$, then the clopen downsets of $X$ are $X_a, a \in L$

11.22 Lemma, from B. A. Davey, H. A. Priestley, Introduction to lattices and order, let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$ and $X_a = \{I \...
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In a distributive lattice, which are the equivalence classes of the projectivity relation on prime intervals?

Let $L$ be a lattice (we can assume that it is distributive), according to Birkhoff (page 72): Two intervals of a lattice are called trasposes when they can be written as $[a \wedge b, a]$ and $[b, a \...
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Need a clarification of the proof that the prime ideal space of a distributive bounded lattice is compact

11.19 Theorem, from B. A. Davey, H. A. Priestley, Introduction to lattices and order, Let $L$ be a bounded distributive lattice, then the prime ideal space $\langle \mathcal{I}_p(L); \tau \rangle$ ...
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Symmetric relations form a CABA

Fix a set X and consider the collection of all symetric relations on it. I also assume that the empty relation is by definiyion symmetric. Well, it is true that the above collection forms a complete ...
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Which are the latticial properties of the quartered aztec diamond?

The aztec diamond is an area of a 2-dimensional square lattice. The quarter aztec diamond it's a part of this area, it can be seen in the following picture: Triangular arrangement of a 2-dimensional ...
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Lattice under a product oder

I have been told that under the product order, {(0,0),(1,0),(0,1),(2,1),(1,2),(2,2)} is not a lattice. I know that a lattice is when joins and meets exist for any pair of elements, and I suspect the ...
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$L$ finite and distributive lattice, then $\mathcal{J}(L)$ (join-irreducible's) is isomorphic, as poset, to $\mathcal{M}(L)$ (meet-irreducible's)

Show that for any finite distributive lattice $L$, $\mathcal{J}(L)$, that is the associated poset of join-irreducible elements of $L$, is isomorphic to $\mathcal{M}(L)$, the associated poset of meet-...
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$P$ poset. $x = \bigvee(\downarrow x\cap U)\Rightarrow \forall x, y \in P$, with $y \lt x$, $\exists a\in U$ s.t. $a \le x $ and $a \nleqslant y$

Le $P$ be a partially ordered set, $U \subseteq P$ and $\downarrow x = \{y \in P : y \le x\}$ (a down set). Show that if $\,\,\forall \,\,x \in P\,\,$ we have $\,\,x = \bigvee(\downarrow x\cap U)\,\,\...
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$I(A)$ and $I(B)$ ideal lattices, then $F(J) = \downarrow \psi(J)$ and $G(U)=\downarrow \phi(U)$ is a connection of Galois between $I(A)$ and $I(B)$.

Let $A$ and $B$ be bounded lattices, $\mathcal{I}(A)$ and $\mathcal{I}(B)$ the ideal lattices of $A$ and $B$ and let $(\phi, \psi)$ be a Galois connection between $A$ and $B$: show that $\forall J \in ...
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