Articles of lattice orders

Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for as a concrete description of these colimits as possible. In particular I hope to do better than just describing them […]

There does not exist an order-preserving surjection from a poset to its down-set lattice?

As in title. By “down-set lattice,” I am specifically referring to the set of lower sets of the poset in question, ordered by set inclusion. I’m not even sure how to start. It feels true, because it looks like a function from a set to a subset of its power set, and I’d be very […]

A construction on boolean lattices is itself a boolean lattice?

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a pair $(\alpha;\beta)$ of functions $\alpha:\mathfrak{A}\rightarrow\mathfrak{B}$, $\beta:\mathfrak{B}\rightarrow\mathfrak{A}$ such that (for every $X\in\mathfrak{A}$, $Y\in\mathfrak{B}$) $$Y\sqcap^{\mathfrak{B}}\alpha(X)\ne\bot^{\mathfrak{B}} \Leftrightarrow X\sqcap^{\mathfrak{A}}\beta(Y)\ne\bot^{\mathfrak{A}}.$$ (Boolean funcoids are a special case of pointfree funcoids as defined in […]

correspondence for universal subalgebras of $U/\vartheta$

Let $U$ be a universal algebra of type $T$, and denote $\mathrm{Con}(U)\!=\!\{\text{congruence relations on }U\}$ and $\mathrm{Sub}(U)\!=\!\{\text{subalgebras of }U\}$. Let “$\leq$” mean “subalgebra”. The correspondence theorem (2.6.20, p. 54) says: $$\mathrm{Con}(U/\vartheta)\!=\!\{\alpha/\vartheta\,;\, \alpha\!\in\!\mathrm{Con}(U),\vartheta\subseteq\!\alpha\},$$ where $\alpha/\vartheta\!=\!\{(a/\vartheta,b/\vartheta)\!\in\!(U/\vartheta)^2;(a,b)\!\in\!\alpha\}$, and also $$(\alpha\wedge\beta)/\vartheta=(\alpha/\vartheta)\wedge(\beta/\vartheta)\;\;\text{ and }\;\; (\alpha\vee\beta)/\vartheta=(\alpha/\vartheta)\vee(\beta/\vartheta).$$ In particular, for a group $G$ and ring $R$ and module $M$ and algebra $A$, […]

A “continuous monoid” is “complete”

Let $A$ be a commutative monoid ordered by $\leq$, that is: $$x\leq y \Rightarrow x + z \leq y +z$$ such that suprema $\bigvee$ of families index by a set $I$ exists and such that: $$\bigvee_{i\in I} (x_i + y) = \bigvee_{i\in I} x_i + y $$ (this seems useful / necessary according to Wikipedia) […]

Is every chain a lattice?

I am asked to prove that every chain is a distributive lattice. Is it true that every chain is a lattice? I am told that 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. I don’t understand how […]

Lattice of continuous functions

Let $\mathcal C [0 , 1]$ be the set of continuous functions from $[0 , 1]$ to the reals. Define $\leq$ on $\mathcal C [0 , 1]$ by $f \leq g \iff f ( a ) \leq g ( a )\; \forall a \in [0 , 1] $. Show that $\leq$ is a partial order […]

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

Problem about Hasse diagrams

Can someone help me to solve this problem. Are these Hasse diagrams lattices?

Coproduct of bounded distributive lattices given as lattices of subsets

Let $X$ be a set. A lattice of subsets of $X$ is a subset of $\mathcal{P}(X)$ containing $\emptyset$ and $X$ and closed under finite intersection and finite union. Such a lattice is therefore a bounded distributive lattice. Let $X_0$ and $X_1$ be sets. Let $\mathcal{L}_0$ be a lattice of subsets of $X_0$ and let $\mathcal{L}_1$ […]