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On the Size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras ∗

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On the Size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras ∗ On the size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras ∗ Peter Freyd [email protected] July 17, 2017 The theory of Heyting Semi-Lattices, Finitely generated hsls are finite. hsls for short, is obtained by adding to the theory of meet-semi-lattices-with-top a The proof takes a while. First, say that binary operation whose values are denoted an hsl is local if there's an element m < 1 x ! y characterized by such that for all x < 1 it is the case that x ≤ m. (The local hsls are the \subdi- x ≤ y ! z iff x ^ y ≤ z rectly irreducibles" of the subject for those who know what that ugly phrase means.) The The easiest equational treatment uses the eponymous example is obtained by starting symmetric version, the double arrow rather with the hsl of open sets of a space, choos- than the single. The binary operation x $ y ing a point, identifying two open sets iff they is characterized by agree when restricted to some neighborhood x ≤ y $ z iff x ^ y = x ^ z of the point. We'll call a local quotient a localization. Each may be defined in terms of the other: Every hsl is embedded into the product of x ! y = x $ (x ^ y) its localizations. x $ y = (x ! y) ^ (y ! x) For two elements a and b we need to find a The $ operation may be equationally localization in which a and b remain distinct. characterized by Define m = a $ b. It suffices to find a local- ization in which m remains different from 1. x $ 1 = x = 1 $ x Reduce by a maximal congruence that does x $ x = 1 just that. In the resulting quotient hsl m has the property that for any non-trivial con- x ^ (y $ z) = x ^ ((x ^ y) $ (x ^ z)) [1] gruence it is the case that m ≡ 1. Let k be Any finite distributive lattice has a unique an element strictly less than 1. Define a con- Heyting structure. Every finite Heyting semi- gruence by x ≡ y iff x ^ k = y ^ k. (The lattice has, of course, a unique lattice struc- third defining equation for $ makes it easy ture and it is necessarily distributive. to see that this a congruence.) Since ≡ is a ∗ This ms was born in 2002. The first appendix was added non-trivial congruence (k ≡ 1) it must be the in 2005, the first footnote in 2013, the subscorings in 2015 and case that m ≡ 1, that is m ^ k = k. In the addendum in 2017. 1[] See 2nd appendix for subscorings. First x and y meet other words, k ≤ m. x $ y in the same way: x^(x $ y) = x^((x^x) $ (x^y)) = x^(x $ (x^y)) = x^((x^1) $ (x^y)) = x^(1 $ y) = x^y and similarly y ^ (x $ y) = x ^ y: Hence for any z ≤ x $ y Assume now that we know that all hsls we have z ^ x = z ^ (x $ y) ^ y = z ^ x ^ y and similarly generated by n elements are finite. Let An be z ^ y = z ^ (x $ y) ^ y = z ^ x ^ y: Finally, if z ^ x = z ^ y then z ^ (x $ y) = z ^ ((z ^ x) $ (z ^ y)) = z ^ 1 = z; that the number of elements in the free hsl on n is, z ≤ x $ y: generators. HEYTING SEMI LAATTICES A local hsl on n + 1 generators has at It's easy to see that the free hsl on one most 1+ An elements. generator has 2 elements. We'll discover be- low that the free on two generators has If m is removed the complement is a sub- hsl 18 elements and is isomorphic the the prod- algebra, because in a local hsl x ! y = m uct 2×3×3 where the numbers name totally only if y = m (and x = 1) and x^y = m only ordered sets. if either x or y is m. One of the n+1 genera- tors must be m. The complement is therefore Define an equationally linear Heyting generated by n of its elements. semi-lattice, or elhsl, to be an hsl that satisfies the further equation: Hence for an hsl on n+1 generators there are only a finite number of isomorphism types ((x ! y) ! z) ^ ((y ! x) ! z)) = z of localizations. From the finiteness of the generating set we know that there are only Any totally ordered set with a top element finitely many maps into any finite hsl. We is a elhsl. And any local elhsl is linear, may conclude that there are only finitely that is, totally ordered (specialize to the case many localizations. Which is quite enough to that z = m). Hence an hsl is equationally force the hsl on n+1 generators to be finite. linear iff it can be embedded in a product of (If the join operation and a bottom are linear hsls. added to the structure, that is, if we con- For a finite set, G, of variables let F (G) sider not hsls but full-fledged Heyting alge- denote the free elhsl on G. F (;) = f1g. bras, then the free ha on one generator is in- Theorem: finite. For a quick proof, partially order the ∼ Y positive integers by taking x <<y iff x+1 < y. F (G) = F (S)? The ha of downdeals is generated by the one- S⊂G; S6=G element downdeal f1g, which is quite enough where we use the standard domain-theoretical for the infinitude of the free ha. Actually, up notation for what is there called the \lifting": to isomorphism, there is only one infinite ha F (S) is the result of adjoining a new bottom on one generator, this one. Indeed, any pair ? element to the poset F (S). of one-generator has of the same cardinality are isomorphic.) Let f(n) be the order of F (n). f(0) = 1. We may read off the following recursion The very first sequence, A000001, in formula for n > 0: Sloane's Encyclopedia Of Integer Sequences 0 1 (http://oeis.org/) counts the number of 2 n 3 n−1 @ A isomorphism types of groups. Both the hsl Y 6 r 7 f(n) = 4(1 + f(r)) 5 and ha analog are the same as his A006982. r=0 That is for X equal to \distributive lattice", \Heyting algebra", or \Heyting The first few terms are: 1, 2, 18, 370386, semi-lattice" the number of isomorphism 143591428101109697973511000185042. It types of X of order n is given by the se- isn't hard to see that there is a constant, C, quence 1, 1, 1, 1, 2, 3, 5, 8, 15, 26, 47, such that f(n) is asymptotically equal to 82, 151, 269, 494, 891, 1639, 2978, 5483, n! 10006, 18428, 33749, 62162, 114083, 210189, C (log2)n : 386292, 711811, 1309475, 2413144, 4442221, 8186962, 15077454, 27789108, 51193086, C is approximately equal to 2.03827. 94357143, 173859936, 320462062, 590555664, For the proofs, note first that for any 1088548290, 2006193418, 3697997558, element k in an hsl, H, the downdeal 6815841849, 12563729268, 23157428823, fx j x ≤ kg has a natural hsl structure 42686759863, 78682454720, 145038561665, and we'll denote that hsl as H=k. The in- 267348052028, 492815778109, ::: clusion map is not a homomorphism but the 2 HEYTING SEMI LAATTICES map from H to H=k that sends x to x ^ k as well call them linear has. (The defini- is. That is, H=k may be viewed as a quotient tion of linear has can be made a bit simpler: structure of H. (Indeed, if H is finite or, more they are those has that satisfy the equation generally, satisfies the descending chain condi- (x ! y) _ (y ! x) = 1.) All of which al- tion, then every quotient structure so arises.) lows us to prove that the free linear Heyting The product decompositions of H are of the algebra on G is constructible as: form H=k1 × H=k2 × · · · where the sequence Y F (S)? fk1; k2;:::g is a \partition of unity", that is, a sequence of elements in H above the bottom S⊆G element whose join is 1 and whose pairwise and that the order of the free linear ha on n meets are each the bottom element. In the fi- generators is: nite case there is a unique largest such. That is, any finite hsl uniquely factors as a prod- (f(n) + 1)f(n) uct of \prime" algebras. (All of this is easy in The first few values are 2, 6, 342, the distributive lattice context. It continues 137186159382. (The same asymptotic expres- to work in the ha context. And even in the sion works as for f(n) with C approximately hsl context.) 4.1545.) Given a finite set, G, let 0 denote the el- V (There is something of a converse of the ement a2G a in F (G) (0 is the bottom el- fact that join is definable in elhsls: if the join ement). For any element x let :x denote operation is definable in a variety of hsls then x ! 0. For each subset S ⊂ G define necessarily it is a variety of equationally linear ^ ^ hsls. See Appendix. There aren't many such tS = ( :a) ^ ( ::a) varieties, by the way. Each is determined by a2GnS a2S the sizes of its local members, that is, each The family ftSgS⊂G; S6=G is a partition of is determined by the smallest n, if any, for unity. To see that, it suffices to show in any Wn which the equation 1 = i=1(xi ! xi+1) is local elhsl generated by G, that there is a satisfied.) unique S ⊂ G such that t names the top el- S The number of isomorphism types of ement and for all other proper subsets of G, elhsls of order n already appears in Sloane t names the bottom element.
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