We have a year of mathematics here (http://www.jahr-der-mathematik.de/) and so we think more often about it. Thus I'd like to ask a question: Consider...
A question for the lattice theorists: is it known if the quasi-variety of balanced lattices with 0 is actually a variety? [Using ^ for meet and v for join, a...
... It is not a variety. If you add a new zero element to any lattice L with zero it becomes balanced, and L is a quotient of this new balanced lattice. Thus,...
... Oh, right. In fact, you even answered my next question, which was about the variety generated by balanced lattices. Yes, of course. I was just being dense....
The message below was sent by Boris Schein. ... Every equivalence relation is a quasi-order (= reflexive and anti-symmetric relation). Every lattice is...
... That's a good one, alright. But since, as I understand it, the quasivariety in question cannot be finitely axiomatized in first order axioms, automated...
OK, here's another one: for each positive integer n, denote by Part (n) the partition lattice of an n-element set (e.g., the lattice of all equivalence...
Friedrich Wehrung
wehrung@...
May 6, 2008 9:38 am
500
Thank you for the lot of input. I am now trying to understand the(se) varieties. JD http://www.cococo.de...
Call A < B a subalgebra gap when A is a maximal proper subalgebra of B. The respective varieties generated by A and B need not form a variety gap V(A) < V(B),...
Vaughan Pratt
pratt@...
May 12, 2008 12:31 am
502
Jonsson¹s lemma is an obvious tool to use here. The varieties generated by the finite Heyting chains are distinct since, by Jobsson, the SIs in Var(n) are...
Brian Davey
B.Davey@...
May 12, 2008 12:37 am
503
Thanks, Brian. Meanwhile I see from Exercise 4.48(18) of MMT that every HA with a gap at the top (1 covering a penultimate b) is SI. So presumably the other...
Vaughan Pratt
pratt@...
May 12, 2008 7:18 am
504
Hi All, This is the answer to the second question by Vaughan: The three-element semigroup S consisting of a two-element left-zero semigroup with additional...
... This is false for the case when A and B are semigroups. Here is an interesting pair. Let B2 = <c,d:cc=dd=0,cdc=c,dcd=d> be the Brandt semigroup of order...
Hello everyone, I have a small problem of pedagogical nature. Is there a simple example of an infinite lattice generated by 3 elements? Of course, free lattice...
Dear Gejza, I try to attach a pdf file of an easily describable, planar, infinite lattice. It is generated by the three elements a,b,c of the picture. If...
Friedrich Wehrung
wehrung@...
May 27, 2008 11:08 am
508
Gejza , The best known easy example is called the herringbone. There is a finite version of it on page 61 of the my text ³Introduction to Lattices and ...
Brian Davey
B.Davey@...
May 27, 2008 11:18 am
509
Gejza & all, Here's an interesting pair of examples: Let A, B, C, and D be the following closed line segments in the real plane: A is the segment with...
... Matt-- I like your example; it makes a nice picture. But I think the statement is backwards somewhere. H is the infinite lattice, since B and C are on...
Hi, I was told that the pdf document (with an infinite 3-generated lattice---actually very close to the "herringbone" mentioned by Brian) had been excluded...
Friedrich Wehrung
wehrung@...
May 28, 2008 8:30 am
513
Yes, this is exacly the type of example that I had in mind. However, (and I hope I am wrong) I think that both H and and K are finite. Let us construct H. The...
Hello David, You are, of course completely correct. I transposed H & K somewhere between my brain and my typed message. :-} Now, I have a response to a PS by...
Thanks Matt, I realised that my PS was unhelpful as soon as I clicked send. :) Brian ____________________________________________ Dr Brian A. Davey Reader and...
Brian Davey
B.Davey@...
May 29, 2008 2:27 am
520
Matt, The fact that the free modular lattice has 28 elements goes back to the very early days of lattice theory: it was proved by Dedekind in 1900: ...
Ralph Freese
ralph@...
May 29, 2008 3:07 am
521
By the way, this reminds me another related question, of which I don't know the answer: Does there exist an infinite, 3-generated lattice, in which every chain...
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