Algebra Univers. 70 (2013) 403409 DOI 10.1007/s00012-013-0258-8Published online September 26, 2013 Springer Basel 2013 Algebra Universalis

The product representation theorem for interlacedpre-bilattices: some historical remarks

Brian A. Davey

Abstract. This note aims to unravel the history of the Product RepresentationTheorem for Interlaced Pre-bilattices. We will see that it has its lattice-theoreticroots in early attempts to solve one of the problems in Birkhos Lattice Theory. Thetheorem was presented in its full generality by Czedli, Huhn and Szabo at a conferencein Szeged, Hungary in 1980 (and published in 1983). This was several years beforeGinsberg introduced bilattices at a conference on articial intelligence in 1986 and inhis foundational paper in 1988.

1. Introduction

This paper is a survey of the history of the Product Representation The-orem for Interlaced Pre-bilattices, and in passing of the closely related resultknown the 90-Degree Lemma. It arose when the author was asked to review apaper on bilattices. After reading the paper, and subsequent reading of relatedpapers on interlaced pre-bilattices, it became clear that the bilattice commu-nity was unaware of some highly relevant papers in the lattice-theory literaturedating back to the 1950s and the 1980s. Unfortunately, the two most relevantlattice-theory papers from the early 1980s on compatible orders of lattices werepublished in conference proceedings and would not have been readily availableto early workers on bilattices. Moreover, some relevant papers from the 1950swere published in Russian, making them inaccessible to many in the West.Nevertheless, with the availability of online resources like MathSciNet and thepower of internet search engines, it is surprising that the intimate connectionbetween interlaced pre-bilattices and compatible lattice orders has not beendiscovered before now. The author hopes that the historical comments set outbelow will help to set the record straight.

Bilattices were introduced by Ginsberg [14, 15] in 1986/1988 as a frame-work for inference in articial intelligence. Work on bilattices continued inthe 1990ssee, for example, Romanowska and Trakul [26], Fitting [12], andAvron [2]. After a period of relative dormancy, there has been a recent spurtof activity on bilattices. Rivieccios PhD thesis [25], which is available online,is a very useful source for the basic theory of (pre-)bilattices. The terminologyconcerning bilattices and their generalisations is not uniform, so we begin by

Presented by G. Gratzer.Received May 20, 2013; accepted in nal form May 22, 2013.2010 Mathematics Subject Classication: Primary: 06B05; Secondary: 03G10.Key words and phrases: bilattice, compatible ordering of a lattice.

404 B. A. Davey Algebra Univers.

xing ours. A pre-bilattice L is simply a set carrying two lattice structures, L1and L2, and therefore possessing four binary operations; to simplify our discus-sion, we shall write L = L1,L2. A bilattice is a pre-bilattice equipped withan involution : L L that is a dual lattice automorphism with respect toL1 and a lattice automorphism with respect to L2. A pre-bilattice is boundedif both L1 and L2 are bounded. A pre-bilattice is called distributive if eachof its four operations distributes over the other three. It is easy to see thatin a distributive pre-bilattice, the lattice operations in each of the lattices areorder-preserving with respect to the order of the other lattice. Pre-bilatticessatisfying this weaker condition are said to be interlaced. Equivalently, L isinterlaced if the order on each lattice is compatible with the operations of theother, that is, the order on each forms a sublattice of the square of the other.It is this view of interlaced pre-bilattices that links them to lattice-theoryliterature from the 1980s.

Every two-factor decomposition of a lattice L1 gives rise to an interlacedpre-bilattice: if : L1 KM is a lattice isomorphism, we dene L2 to be theuniquely dened lattice on the underlying set of L1 such that : L2 KMis a lattice isomorphism. When this happens, we will say that the lattice L2and the pre-bilattice L = L1,L2 arise from a two-factor decomposition of L1.We intend to trace the history of the following fundamental result.

Product Representation Theorem for Interlaced Pre-bilattices. Apre-bilattice L = L1,L2 is interlaced if and only if it arises from a two-factordecomposition of L1.

2. The lattice history

We will see that construction of a second lattice structure L2 on the samebase set as a lattice L1 via a two-factor decomposition of L1 has its roots inthe 1950s in attempts to solve a problem from the second edition of BirkhosLattice Theory [5]. Problem 8, which is repeated as Problem 6 in the thirdedition [6], states: Find a necessary and sucient condition on a lattice L,in order that every lattice M whose (unoriented) graph is isomorphic with thegraph of L be lattice-isomorphic with L. Here graph-automorphism of L meansan automorphism of the undirected covering graph G(L) of L. In approachingthis problem, it is natural to consider the following more general problem:Given a nite lattice L1, describe all lattices L2 such that G(L2) is isomorphicto G(L1). It is trivial that if L2 arises from a two-factor decomposition of L1,then G(L2) is identical to G(L1). Surprisingly, for a large class of lattices, thisis essentially the only way that such pairs of lattices can arise. This problemwas studied intensively, especially by J. Jakubk, between the mid 1950s andthe mid 1980s.

In 1980, attention turned from considering pairs of lattices with the sameunoriented graph to pairs of lattices with compatible orders, that is, in the

Bilattices: some historical remarks 405

modern parlance, to interlaced pre-bilattices. The full statement and proofof the Product Representation Theorem for Interlaced Pre-bilattices was pre-sented by Czedli, Huhn and Szabo at a conference in Szeged, Hungary in 1980and appeared in print in 1983. They also gave the rst proof of the 90-DegreeLemma. In the timeline below, statements from the older literature have beentranslated into modern bilattice terminology.

1947 Kiss self-published text [20] contains many examples (but no theory) ofmultiple lattice structures on a product of lattices obtained by reversingthe order in one or more coordinates. (See also Birkho and Kiss [7] andBirkhos review of Kiss text [4].) On page 106, Kiss gives the exampleof 2 2 with four possible orders. The bilattice on 2 2 is considered inthe context of Boolean algebras on pages 136139. Kiss discussion of alogic based on the four-element bilattice on pages 300303 is a precursorto Belnaps four-valued logic [3].

1951 Arnold [1] considered algebras L = L;,, such that L;, isa lattice, L; is a semilattice and all possible distributive laws holdamongst , and . The Product Representation Theorem for (notnecessarily bounded) distributive pre-bilattices follows immediately fromhis Theorem 17. Among his preliminary calculations (Theorem 2) is thefact that every distributive pre-bilattice is interlaced.

1954 Jakubk and Kolibiar [16] also proved the Product Representation The-orem for (not necessarily bounded) distributive pre-bilattices. In addi-tion, they proved that, for nite distributive lattices L1 and L2 with thesame underlying set, G(L1) = G(L2) if and only if L := L1,L2 is adistributive pre-bilattice.

1954 Jakubk [17] proved that, for nite modular lattices L1 and L2 with thesame underlying set, G(L1) = G(L2) if and only if L2 arises from atwo-factor decomposition of L1.

1975 Jakubk [18] extended his 1954 result to discrete modular lattices, thatis, those in which each bounded chain is nite. He proved that if L1 is adiscrete modular lattice and L2 is a discrete lattice such that G(L1) =G(L2) then L2 arises from a two-factor decomposition of L1.

1980 Czedli, Huhn and Szabo [10] considered compatible lattice orders oflattices and proved both the 90-degree Lemma, see (c) below, and theProduct Representation Theorem for Interlaced Pre-bilattices. Theirmain results are as follows.(a) A lattice that has a compatible bounded order is itself bounded.(b) A bounded compatible order of a lattice is a lattice order.(c) The compatible lattice orders on a bounded lattice are in one-to-one

correspondence with the complemented neutral elements of L, andconsequently every such order arises from a two-factor decomposi-tion of L. This appears to be the rst proof of what is now knownas the 90-degree Lemma (see Jung and Moshier [19, page 31]).

406 B. A. Davey Algebra Univers.

The proof is straightforward and proceeds via a direct calculationof mutually inverse maps between the set of complemented neutralelements of L and the set of compatible lattice orders of L.

(d) Every compatible lattice order of a lattice L arises from a two-factor decomposition of L. The Product Representation Theoremfor Interlaced Pre-bilattices follows immediately from this result,since there compatibility is assumed in both directions.The proof uses the compatible order to dene a pair of factor con-gruences 1, 2 on L such that L = L/1 L/2. The descriptionof 1 and 2 depends on the fact that the two orders on L havethe same closed intervals, which in turn depends upon the boundedversion of the theorem proved in (c).

1983 Kolibiar [21] proved that compatible orders on a lattice L are in a one-to-one correspondence with two-factor subdirect representations of L. Thedescriptions of the congruences 1, 2 on L required for the subdirectrepresentation is much simpler that the one given by Czedli, Huhn andSzabo [10]: if denotes the natural order on the lattice L and denotesa compatible order on L, then

x 1 y (v L)[x v y & x v y],

x 2 y (u L)[x u y & x u y].

Moreover, he proved that a compatible order of a lattice is itself a latticeorder if and only if it is both up- and down-directed. This gives theCzedli, Huhn and Szabo result (b) above as an immediate corollary.

1984 Rosenberg and Schweigert [27] consider compatible orderings of algebraswith a majority term, of semilattices and of lattices. They indepen-dently obtain the results of Kolibiar [21] stated above via very similarproofs. They use their results to give a new proof of Czedli, Huhn andSzabos result that every compatible lattice order on a lattice arises froma two-factor decomposition of L. They also made the observation thatcompatibility of lattice orders is symmetric: see (1) below.

1995 Czedli and Szabo [11] give a particularly straightforward proof that thelattice Quord(L) of quasiorders on a lattice L is isomorphic to Con2(L),the square of the congruence lattice of L. The mutually inverse isomor-phisms L : Quord(L) Con2(L) and L : Con2(L) Quord(L) aregiven by

(( ) ( ), ( ) ( )), and(1, 2) (1 ) (2 ).

Under this isomorphism compatible orders on L correspond to pairs(1, 2) Con2(L) with 1 2 = 0, that is, to subdirect represen-tations of L, and compatible lattice orders on L correspond to pairs(1, 2) Con2(L) with 1 2 = 0 and 1 2 = 1, that is, to two-factordecompositions of L.

Bilattices: some historical remarks 407

We trace the bilattice history of the Product Representation Theorem forInterlaced Pre-bilattices in the next section. Before doing so, we present severalimmediate consequences for pre-bilattices of the fact that every compatiblelattice order of a lattice L arises from a two-factor decomposition of L. Mostof these statements are well known,

(1) If L2 is a compatible lattice order of a lattice L1, then L1 is a compati-ble lattice order of L2 (thus the denition of interlaced pre-bilattice canbe weakened). This was proved independently in the bounded case byPynko [24].

(2) A pre-bilattice L = L1,L2 is distributive if and only if it is interlacedand either L1 or L2 is a distributive lattice.

(3) If L = L1,L2 is an interlaced pre-bilattice, then L1 satises a latticeidentity if and only if L2 does.

(4) Since lattices have factorizable congruences (see Fraser and Horn [13]) thecongruences on M K are precisely the products of congruences on thefactors, from which it follows at once that an equivalence relation on theunderlying set of an interlaced pre-bilattice L = L1,L2 is a congruenceon L if and only if it is a congruence on either L1 or L2.

(5) An interlaced pre-bilattice L = L1,L2 is subdirectly irreducible if andonly if L1 is a subdirectly irreducible lattice and L2 {L1,L1}.

(6) If L2 is a compatible lattice order of a lattice L1, then L1 and L2 sharethe same closed intervals (as sets), and consequently, if L = L1,L2 is aninterlaced pre-bilattice, then L1 and L2 have the same unoriented coveringgraph.

3. The bilattice history

The bilattice community were not aware of the literature on compatibleorderings of lattices and reproved the Product Representation Theorem forInterlaced Pre-bilattices over a number of years. As Mobasher, Pigozzi, Slut-ski and Voutsadakis remark, This theorem has a complicated evolution [22,page 113].

1989 Romanowska and Trakul [26] proved the Product Representation Theoremin the case of bounded interlaced bilattices (in which case L1 = L2), butthe proof is easily modied to yield the proof for bounded interlaced pre-bilattices. The proof is somewhat indirect and uses Plonka sums.

1990 M. Fitting [12] proved the Product Representation Theorem in the caseof complete distributive interlaced pre-bilattices.

1996 A. Avron [2], independently of Romanowska and Trakul, proved the Prod-uct Representation Theorem for bounded interlaced pre-bilattices. Theproof is short and direct and avoids the machinery of Plonka sums usedin [26]. The 90-degree Lemma is proved here.

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2000 Pynko [24] independently obtained the extension to bounded interlacedpre-bilattices, which he referred to as regular bilattices. He also showedthat, in the bounded case, it was enough to assume only that the orderon L2 is compatible with the order on L1.

2000 Mobasher, Pigozzi, Slutski and Voutsadakis [22] also independently provedthe extension to bounded interlaced pre-bilattices. In a footnote, theythank a referee for pointing out the earlier work of Avron [2] and the al-most simultaneous work of Pynko [24]. They show that the representationis in fact the object half of a category equivalence between the category ofbounded interlaced bilattices and the categorical square of the categoryof bounded lattices.

2006 Movsisyan, Romanowska and Smith [23] extended the Product Represen-tation Theorem to the not-necessarily bounded case. The proof is basedon the 1989 proof given for bounded interlaced bilattices in [26] and souses Plonka sums.

2010 As part of his PhD thesis, Rivieccio [25] also independently extended theProduct Representation Theorem to the not-necessarily bounded case.The proof is direct and proceeds by proving that

1 := (2 1) (2 1)1 and 2 := (2 1) (2 1)1

are congruences on L that yield the required factorisation. These con-gruences agree with those dened in Kolibiar [21] and Rosenberg andSchweigert [27]. The proof was published a year later by Bou and Riviec-cio [9].

2011 With the full Product Representation Theorem available to them, Bou,Jansana, and Rivieccio [8] observe that the category of interlaced pre-bilattices is equivalent to the the categorical square of the category oflattices. This extends the corresponding result in the bounded case byMobasher, Pigozzi, Slutski and Voutsadakis [22].

Acknowledgement. The author would like to thank Hilary Priestley forhelpful comments on several drafts of this paper.

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[13] Fraser, G.A., Horn, A.: Congruence relations in direct products. Proc. Amer. Math.Soc. 26, 390394 (1970)

[14] Ginsberg, M. L.: Multi-valued logics, in: Proceedings of AAAI-86, Fifth NationalConference on Articial Intelligence, pp. 243247. Morgan Kaufmann Publishers, LosAltos (1986)

[15] Ginsberg, M. L.: Multivalued logics: A uniform approach to inference in articialintelligence. Comput. Intelligence 4, 265316 (1988)

[16] Jakubk, J., Kolibiar, M.: On some properties of a pair of lattices. Czech. Math. J. 4,127 (1975) (Russian with English summary)

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[19] Jung, A., Moshier, M.A.: On the bitopological nature of Stone duality. TechnicalReport CSR-06-13, School of Computer Science, University of Birmingham (2006)

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Brian A. Davey

Mathematics and Statistics, La Trobe University, Victoria 3086, Australiae-mail : B.Davey@latrobe.edu.au

The product representation theorem for interlacedpre-bilattices: some historical remarksAbstract1. Introduction2. The lattice history3. The bilattice historyAcknowledgementReferences