0
votes
0answers
37 views

Is it true that the set of points minimizing their distance to a multiset of intervals from a distributive lattice is an interval?

Let $(E, \preceq)$ be a finite distributive lattice, $H_E$ be the Hasse diagram of $E$ and $d$ be the distance on $E \times E$ defined as the length of the shortest path in $H_E$ between any pair of ...
1
vote
1answer
98 views

generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
0
votes
0answers
121 views

Is the direct product of distributive inclusions of groups, modular?

Let $H$ a subgroup of $G$ and $\mathcal{L}(H \subset G)$ the lattice of intermediate subgroups ($\mathcal{L}( G)$ if $H= \{ e \}$). Definitions: A lattice $(L, \wedge, \vee)$ is : - Distributive if ...
0
votes
2answers
125 views

Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
2
votes
0answers
170 views

Existence of inclusions of finite groups with a particular lattice property

Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by : $(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
20
votes
7answers
1k views

Was lattice theory central to mid-20th century mathematics?

Four years ago, I read a book on the history of mathematics up to 1970 or so. It was very interesting up until the end. The last few chapters, though, were on lattices. The author claimed that ...
2
votes
2answers
279 views

Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra but have received no answer. Sorry, I ask a ...
2
votes
2answers
122 views

Are there atomistic ortholattices which are not modular?

Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...
1
vote
0answers
57 views

Reference request: Compact metrizable semilattices with small connected semilattices are absolute retracts

Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path connected. I need a reference for a proof that $X$ is an absolute retract. Here is ...
5
votes
1answer
207 views

Complete anti-chain lattices and the axiom of choice

Hello, everyone. I'm trying to find out about lattices of anti-chains, and was wondering whether you could help me with getting to grips with a Comp. Sci. paper I'm struggling with. I've been reading ...
1
vote
0answers
56 views

variant on ring objects in the category of complete lattices

Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a ...
2
votes
1answer
101 views

Is the complete lattice of inflators on a frame a frame?

Yup, it may sound like an inocent question to many of you; but a very good friend of mine is completely baffled in his research about lattices of inflators on a frame. He asked me very kindly to post ...
2
votes
1answer
273 views

When are the join-irreducibles in a complete lattice join-dense?

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee R|R\subseteq A\}$. An element $a\in L$ is said to be join-irreducible if $a\neq 0$ and if $a=x\vee y$ then $a=x$ or ...