Date: Thu, 26 Apr 2012 09:44:08 +0100 (BST) From: "Prof. Peter Johnstone" To: Mark Jason Dominus Subject: Re: How many Heyting algebras are there with n elements? In-Reply-To: <10302.1335367059@plover.com> Message-ID: References: <10302.1335367059@plover.com> Dear Mark, No, I haven't worked on this question. (I did once consider the question "how many elements are there in the free Heyting semilattice on n generators?", which may be what Robin was thinking of.) However, since every finite distributive lattice is a Heyting algebra, your question is equivalent to "How many distributive lattices are there with n elements?", and I'm sure there must be work on this in the literature -- though I can't provide specific references. Best regards, Peter Johnstone Message-ID: <20120426101100.182633jdfgz1gwvo@webmail.seas.upenn.edu> Date: Thu, 26 Apr 2012 10:11:00 -0400 From: pjf@seas.upenn.edu To: Mark Jason Dominus Subject: Re: Number of small Heyting algebras References: <10203.1335367015@plover.com> In-Reply-To: <10203.1335367015@plover.com> Every finite distributive lattice has a unique Heyting-algebra structure. The number of such of order n is -- the last I've heard -- still considered by many to be among the most important questions in combinatorics. Just google "number of distributive lattices" to find some exponential bounds. I had totally forgotten "On the size of Heyting Semi-Lattices" from 10 years ago. Thanks for reminding me! I found the ms and it's now available at http://www.math.upenn.edu/~pjf/Heyting.pdf Best thoughts, Peter Freyd