UNIVERSAL ALGEBRA AND LATTICE THEORY: A STORY AND THREE RESEARCH PROBLEMS

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1 Universal Algebra and its Links with Logic, Algebra, Combinatorics, and Computer Science Proc. "25. Arbeitstagung liber Allgemeine Algebra", Darmstadt 1983 P. Burmeister et al. (eds.) Copyright Heldermann Verlag UNIVERSAL ALGEBRA AND LATTICE THEORY: A STORY AND THREE RESEARCH PROBLEMS G. Gratzer I would like to tell a story and present three researoh problems on universal algebra and lattioe theory, with speoial emphasis on the interaotion of these two fields. Although the topics chosen reflect my special interests, I hope they give the reader the flavour of this area of research. The story: A free m-latt1ge. Lattice theory is two faceted: on the one hand, a lattice is a universal algebra <L; A, V) with two binary operations satisfying eight identities (see, e.g. [8], I.1); on the other hand, a lattice is a special kind of poset <L; S) in which any two elements have a least upper bound and greatest lower bound. The universal algebraic approach gives lattice theory such concepts as congruence relation, free lattice, free lattice on a poset P (notation: F(P», free prod~ct, equational class, etc. The completeness. poset approach gives for example the concept of The two approaches are sometimes difficult to reconcile. For instance, there is no free complete lattice of 3 generators since there are complete lattices of arbitrarily large size completely generated by 3 elements. In a series of papers [10], [11], [12], [13], [14], [15], D. Kelly and the author continued the development of the theory of m-lattices which was started by P. Crawley and R. A. Dean [2]: a poset L is an m-lattice if all subsets X with o < Ix I < m have a. least upper bound and greatest lower bound (see also [9]). The theory of m-lattices is an important contribution of universal algebra to lattice theory. survey of some aspects of this theory, see [16].) (For a

2 2 G. Gratzer The development of this theory is quite lattice theoretic in nature. However, occasionally, universal algebra comes to the rescue. Let H be the poset of Figure 1. The free m-lattice D(m) (with an additional a and 1) on H is shown of Figure 4. It is made up of the lattice A of Figure 2, the "mirror image" of A: the lattice B, for each 1 a dyadic rational, a copy C1 0 f the I a t tice C( m) 0 f FiguI'e 2 ( C( tt o ) i sin the "middle" of C(m); the upper part of c(m) is not shown), and for each real t, 0 < t < 1, t non-dyadic, a copy C i of the two-element chain. if it is one-to-one, then It is an isomorphism iff case, 58 is isomorphic to image of B. Figure 5 shows in more detail how the elements of A and B interact with the Ct' The result D(N o ) = F(9) is due to I. Rival and R. Wille (27J. The proof that D(m) - {y, Y'} is the free nt-lattice on H uses a result of P. Crawley and R. A. Dean [2]; it is necessary to verify condition (W m ): if P = I\x ~ VY = q, then (X u Y) n [p, q] ". is. ( X, Y.& D ( m), 0 < I X I,!Y I < m}. The proof is very long and requires a detailed analysis of where p and q are. In the paper [13], D. Kelly and the author found a universal algebraic proof avoiding most of the tedious computations. This proof uses the m = NO case (the result of Rival and Wille) and some universal algebraic trivialities. Let! be a variety (equational class) of algebras of some finitary or infinitary type. For ~ = (A; F> E ~ and 9 A, we define the partial K-algebra ~ = (H; F> on 9 as ' a" ) = a E H follows: if f E F, a O ' a 1,... E 9 and f(a o in ~, then (and only then) f( a ' a" ) is defined on H O and equals a. ~ is called a relatiye. algebra of ~. If e = (B; F> is a partial algebra of the type of K, then F(~) denotes the free (-algebra generated by I. The canonical map of 5'8 into F (~) is not necessarily one... to-one; it is an embedding of ~ into F( 58). 58 is a partial ~-algebra; in this the relative algebra of ~ on the Now let u o and ~1 be partial ~-algebras,

3 Universal algebra and lattice theory A O n A 1 = A 2 suoh that ~2 is the same as a relative algebra of ~O and of ~1 We shall say that ~O and ~1 can be etrongly amalgamated oyer ~2' if there is an algebra ~3 ~ ~ of whioh both ~O and ~1 are relative algebras and A O n A 1 = A 2 in ~3. The following easy lemma should give the flavour of the new proof e..m.mA. Let U be a partial (-algebra, let At $ A, and let ~t be the oorresponding relative algebra of ~. If ~ and F(~t) can be strongly amalgamated over Ut, then the subof F(~) generated by At is algebra [At) naturally isomorphic to F ( U t ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * First Problem: Let U be an idempotent algebra such that Pi (tt) < a::i for all i. Does there exist an n suoh that <P2(~)', Pn(U» has the Minimal Extension Property? The concept of the Pn-sequence was introduoed by E. Marozewski. Take an algebra ~,and consider the sequence <f o ' f 1, f 2, >, where f n is the size of the free algebra over ~ with n generators. The Basic Problem is the characterization of such sequenoes. Now look at Figure 6, depicting the free algebra with 2 generators. The shaded areas are two copies of F(l), they overlap in F(O). Thus f 2 ~ 2f, - f O There is a similar inequality for f n for every n (the, so oalled, inolusion exolusion prinoiple). So it is more convenient to study Pn' the number of essentially n-ary polynomials ("essentially" means that they depend on all n variables). The size of the unshaded area in Figure 6 is P2. So let PO(U) be the number of oonstant unary polynomials, let Pl(~) be the number of non-oonstant (essentially unary) polynomials exoluding p(x) = x; for n > 1, P (ti) is n the number of essentially n-ary polynomials. The Basio Problem restated is: which sequenoes <Pn> oan be represented as <Pn(~» for some algebra U. It was proved in [22] that if Po - 0, <Pn> is always so representable. In the oase Po = 0, there are many representation

4 4 G. Gratzer results. The tollowing, also from (22], is typical: If > 0 for all n ~ 1, or if n divides for n even and > 0 for n odd, or if n divides for all n, then is so repre.sent.able. All three oonditions impose some restriction on each Pn' but the ~ of Pi I,. j. does not influenoe the size of if In the idempotent case = P, = 0, that is, there are no nullary operatioas and f(x,, x) = x for all operations f) the situation is very different ([21] aad [24]): Let U be aa idempotent algebra which is not equivalent to a semilattiae, a diagooal algebra «B o ; t>, where 1 1 f«a 1,..., a>,,... J 1 0 <a 1,,, defines the o-ary operation f), an idempotent reduat of a Boolean group (let <G; +> be a Boolean group, that is, 2x = 0; the idempotent reduat is <0; x + y + z». Then there exists an n such that (~O < eu) < So in the idempotent case < is strictly iriereasing trom some point on (with three exceptions). Of course, given a eu», we oan trivially ooastruct a <PiCe» whioh agrees with (U» up to a, and from n on <ptce» increases faster: (m> > (U) for all i > n. For the free idempoteat semigroup satisfying xyz = xzy, obviously P n = n for all n ~ 2. The following result of J. P~onka [26] proves an astonishing converse: Let U be an idempotent algebra satisfying (U) = 2, (U) = 3, and P4(U> = 4. Then p eu) ~ n for all n a 2. n In other words, <0, 0, 2, 3, 4) has a "minimal" extension <0, 0, 2, 3, 4, 5, n ); every other extension is a the minimal extension. Definition: <0, 0, P > has the Minimal Extension Property Iff there ' a~ 'lctembotent algebra.' satisfying (1) Pi = Pi (U) for 2 Sis nj (Ii) for any Idempotent algebra 8. it Pi = Pi(8) for 2 SiS D, then Pj(t{) S (8) tor all jan.

5 Universal algebra and lattice theory It was proved in my paper with R. Padmanabhan [17] that (0, 0, 1, 3, 5> has the Minimal Extension Property and the minimal extension is given by the algebra <Ai 0>, where (Ai +> is an abelian group of exponent 3, and x 0 y = 2x + 2y. The First Problem asks, whether any idempotent algebra ti gives rise to a sequenoe <0, 0, Pi(ti),, Pn(ti» with the Minimal Extension Property if n is large enough. For a more detailed introduotion to this problem, see (4]. For more reoent referenoes, oonsult Algebra Universalis, espeoially, A. Kisielewioz [24]. Second Froblem: Find all lattioes K with ~he 'property that whenever K oan be embedded in the ideal lattioe I(L) of a lattioe L, then K oan also be embedded into L. 5 This problem illustrates how developments in universal algebra open up new fields in lattioe theory. Chapter 7 of (7] is devoted to the following question in universal algebra: whioh first ordered properties are preserved under algebraio oonstruotions (e.g., formations of subalgebras, homorphio images, direot produots). The formation of the ideal lattioe I(L) of a lattioe L is an algebraio oonstruotion that is speoial to lattioe theory. So it is natural to raise the question what properties are pr~served under the formation of ideal lattices. I discussed this question briefly in 1961 (see (5]). Sinoe there seem to be too many such properties, I proposed to investigate the special case: the lattioe has a sublattice isomorphio to a given lattioe K. This is how we arrive at the Second Problem. Lattices K satisfying this property are called transferable. There is also a stronger oonoept. Let <p be an embedding of K into I(L)i for every a E K, a<p is an ideal of L. If K is transferable, then there is an embedding ~ of K into L. Now it seems natural to require, for a E K, that a~ be in the ideal a<p, but not in any b<p where b < a. If there is always such a map ~, K is oalled aharply transferable.

6 6 G. Gratzer It is an instructive exercise to check that N 5 is sharply transferable. (In fact, K. A. Baker and A. Hales observed that all finite projeative lattices are transferable; Algebra Universal Is 4 (1974), ) Sharp transferability is easier to handle than transferability since we know roughly where aw should be. Let us start the discussion with some definitions: 1. Let <PI ~> be a partially ordered set. For X t I S P, define X < Y to hold if and only if for every x X there exists y E I suoh that x ~ y. 2. Let <S; v> be a join-semilattice, p ~ S, and J S S. We say that <p, J) is a minimal pair if and only if the following three conditions hold: (i) p ~ J; (ii) p ~ V; (Iii) if J'.s S, P ~ VJ', a1?d J' < J,then J.= J' 3. A semilattioe (5; v> is said to satisfy (T) if and only if there exists a linear order relation R on S suoh that if <p, J> is a minimal pair, then p R x holds for all X E J. 4. A lattice <L; At v> is said to satisfy ( if and only if <L; v> satisfies (T), and to satisfy ) if and only if the dual of <L; A> satisfies (T). Iheorem. A finite lattice is sharply transferable if and only if it satisfies the three oonditions (Tv), (T A ), and on. Condition (N) is (N m ) the condition: x A y S U X A y, U v v] ~ ~. of "The story" with m = ~O' that Is, v v implies that {x, y, u, v} n Later, in 19], I generalized this result with Platt to arbitrary lattices. The first result for transferability is in [5] (see [6] for a proof): A transferable lattice cannot have doubly reduoible elements. The proof relies on the following result: Every lattice can be embedded in the ideal lattice of a lattice without doubly irreducible elements.

7 Universal algebra and lattice theory This led to the following 7 Conjecture. The class of transferable lattices is the intersection of all classes K of lattices with the property that every lattice can be embedded in the ideal lattice of some lattice in ~. Such classes were investigated in [18] and [20]. Sufficiently many such classes were constructed to conclude: Theorem. For finite lattices, transferability and sharp transferability are equivalent. It is quite likely that proving the Conjecture would lead to a solution of the Second Problem. * * * * * * * * * * * * * * * * * * * * * * * * * * Third fcoblem: Find a nontrivial class! of groupoids (algebras with one binary operation +) such that for every ~ E K, a, b, c, d E A, c.i. d (9(a, b» iff c = a + y and d = b + Y for some YEA. This problem comes from a universal algebraic problem x1a lattice theory. For an algebra ~,a, b E A, let 9(a, b) denote the smallest congruence relation under which a.i. b. Mal'cev's Lemma (see, e.g., (7], Theorem 10.3) describes 9(a, b) as follows: c.i. d (9(a, b) iff there exists a sequence Zo = c, zl', zn = d of elements of A such that for each i < n, there exists a polynomial P.(x, Xl', x) with 1 m Pi (a, Yl'"'' Ym) = zi+f(i)' Pi (b, Y1'..., Ym) = zi+l-f(i) for some Yl"'" Ym E A, where f is a choice function, f(i) = 0 or f(1) = 1- The choice of n, of PO"'" P n - 1 ' and of f depends on a, b, c, and d. Would it not be nice, if they could be chosen 1ndependently of a, b, c, d? In most classical algebraic systems this is not the case. Take a Boolean group: c _ d (9(a, b» Iff c + d = a + b or c + d = 0; 1n th1s case, n = 1 t p(x, y) ;: x + y, but f depends on a, b, 0, and d.

8 8 G. Gratzer For an interesting example, lattice theory came to the rescue. I proved with E. T. Schmidt [23J that in a distributive lattice L, a, b, c, d to L, a S b, c S d, c.:. d (a v Y1) " Y2 = c, (b v Y1) " Y2 = d for some there are examples where PO', Pn-1' fare whole equational class...., Pn-1 ' Po' and f is called a congruence scheme E. An equational class ~ is said to have a uniform congruence scheme,e, if for all ~ to K, a, b, c, d to A, c _ d ('l (a, b.)) can always be described with the same PO', This concept was introduced by R. Magari [25J. In [1J, I ('l(a, b)) iff Yl' Y2 to L. Thus the same for a Pn-1 ' and f. See also [3]. considered with J. Berman the question how congruence schemes may look like. If there are constants, the problem seems very difficult. Even n = 1, f(o) = 0, p(x, y) = o + «0 + x) + y) cannot be the congruence scheme for an algebra with more than one element. Now if there are no constant operations, then the result is very nice: Theorem. PO', Pn-1' f is the uniform congruence scheme for a nontrivial variety iff all Pi are at least binary. The condition, all Pi are binary, simply means that no Pi has only x as variable. The unspecified type. equational class constructed in the proof is of Of course, the type contains all the operations needed to build up the Pi' but it contains a number of additional operations. The Third Problem asks, in the simplest possible case, what happens if the type is fixed. The congruence scheme in the Third Problem is p(x, y) = x + Y (and f(o) = 0). One can raise the same problem with any other reasonable polynomial (y + (x + y), Y + «x + y) + x), etc.) or pairs of polynomials, e.g., PO(x, y) = x + y, p,(x, y) = y + «x + y) + (x + y» f(o) = 0, f(1) = 1 that is, iff c _ d ('l(a, b)) c = a + Y1' b + Y1 = Y2 + «a + Y2) + (a + Y2)' d = Y2 + «b + Y2) + (b + Y2))

9 Universal algebra and lattice theory References [1] J. Berman and G. Gratzer: Uniform representations of congruence schemes. Pacific J. Math. 76 (1978), [2] P. Crawley and R. A. Dean: Free lattices with infinite operations. Trans. Amer. Math. Soc. 92 (1959), [3] E. Fried, G. Gratzer, and R. W. Quackenbush: Uniform congruence schemes. Algebra Universalis 10 (1980), [4] G. Gratzer: Composition of functions. In Proceedings of the Conference on Universal Algebra, Queen's University, Kingston, Onto (1969), [5] G. Gratzer: Universal Algebra. In Current Trends in Lattice Theory. D. Van Nostrand, (1970), [6] G. Gratzer: A property of transferable lattices. Proc. Amer. Math. Soc. 43 (1974), [7] G. Gratzer: Universal Algebra. Second Edition. Springer Verlag, New York; Heidelberg, Berlin, [8] G. Gratzer: General Lattice Theory. Pure and Applied Mathematics Series, Academic Press, New York, N. Y.; Mathematische Reihe, Band 52, Birkhauser Verlag, Basel; Akademie Verlag, Berlin, (Russian translation: MIR PUblishers, Moscow, 1982.) [9] G. Gratzer, A. Hajnal, and D. Kelly: Chain conditions in free products of lattices with infinitary operations. Pacific J. Math. 83 (1979), [10] G. Gratzer and D. Kelly: A normal form theorem for lattices completely generated by a subset. Proc. Amer. Math. Soc. 67 (1977), [11] G. Gratzer and D. Kelly: Freem-products of lattices. 1 and II. Colloq. Math., to appear. [12] G. Gratzer and D. Kelly: The free m-lattice on the poset H. ORDER, to appear. [13] G. Gratzer and D. Kelly: A technique to generate m-ary free lattices from finitary ones. [14] G. Gratzer and D. Kelly: An embedding theorem for free m-lattlces on slender posets. [15] G. Gratzer and D. Kelly: A description of free m-lattices on slender posets.

10 log.gratzer [16] G. Gr~tzer and D. Kelly: The construction of some free m-lattices on posets. Proceedings of the Lyon Conference on Ordered Sets. [17] G. Gratzer and R. Padmanabhan: and non-associative groupoids. (1971), On idempotent, oommutative, Proc. Amer. Math. Sao. 28 [18] G. Gritzer and C. R. Platt: Two embedding theorems for lattices. Proc. Amer. Math. Soc. 69 (1978), [19] G. Gritzer and C. R. Platt: A characterization of sharply transferable lattices. Canad. J. Math. 32 (1980), [20] G. Gritzer, C. R. Platt, and B. Sands: Embedding lattices o lattice of ideals. Paoific J. Math. 85 (1979), -75. [21 G. Gritzer and J. Plonka: On the number of polynomials of an idempotent algebra. I and II. Pacific J. Math. 32 (1970), and 47 (1913), [22] G. Oritzer, J. Plonka, and A. Sekanina: polynomials of a universal algebra. I. (1910),9-11. On the number of Colloq. Math. 22 [23] G. Gr~tzer and E. T. Schmidt: Ideals and congruence relations in lattices. Acta Math. Acad. Sci. Hungar. 9 ( 1958 ), [24) A. Kisielewicz: The sequences of idempotent algebras are strictly increasing. Algebra UniversalIs 13 (1981), [25] R. Magari: The classification of idealizable varieties (Congruenze ideali IV), J. of Alg., 26 (1973), [26] J. P~onka: On algebras with at most n distinct entially n-ary operations. Algebra Universalis 1971) [27] I. Rivai and R. Wille: Lattices freely generated by ially ordered sets: Which can be "drawn"? J. Reine. Math. 310 (1979), University of Manitoba Winnipeg, Manitoba R3T 2N2 Canada

11 Universal algebra and lattice theory 11 The poset H Figure I L bi+1 OJ LI i limit 1 L o Q"'Oo The lottice A Figure 3 Figure 2

12 G. Gratzer t real nondyadic <0.1> <1,0> i dyadic The m-lofticed(m) Figure 4 A <s.r> 8 r< t<s r. s dyadic t real noodyadic Details of D(m) Figure 5

13 Universal algebra and lattice theory 13 Figure 6