International audienceThe study of partial clones on 2 := {0, 1} was initiated by R. V. Freivald. In his fundamental paper published in 1966, Freivald showed, among otherthings, that the set of all monotone partial functions and the set of all self-dual partial functions are both maximal partial clones on 2. Several papers dealing with intersections of maximal partial clones on 2 have appeared after Freivald work. It is known that there are infinitely many partial clones that contain the set of all monotone self-dual partial functions on 2, and the problem of describing them all was posed by some authors. In this paper we show that the set of partial clones that contain all monotone self-dual partial functions is of continuum cardinality on 2

Couceiro, Miguel

Haddad, Lucien

Rosenberg, Ivo G.

arXiv

Rosenberg, Ivo, G.

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HAL Id: hal-01093942https://hal.archives-ouvertes.fr/hal-01093942Submitted on 6 Aug 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Partial clones containing all Boolean monotone self-dualpartial functionsMiguel Couceiro, Lucien Haddad, Ivo G. RosenbergTo cite this version:Miguel Couceiro, Lucien Haddad, Ivo G. Rosenberg. Partial clones containing all Boolean monotoneself-dual partial functions. Journal of Multiple-Valued Logic and Soft Computing, Old City Publishing,2016, 27 (2-3), pp.183-192. ￿hal-01093942￿PARTIAL CLONES CONTAINING ALL BOOLEANMONOTONE SELF-DUAL PARTIAL FUNCTIONSMIGUEL COUCEIRO, LUCIEN HADDAD, AND IVO G. ROSENBERGAbstract. The study of partial clones on 2 := {0, 1} was initiated byR. V. Freivald. In his fundamental paper published in 1966, Freivaldshowed, among other things, that the set of all monotone partial func-tions and the set of all self-dual partial functions are both maximalpartial clones on 2.Several papers dealing with intersections of maximal partial cloneson 2 have appeared after Freivald work. It is known that there areinfinitely many partial clones that contain the set of all monotone self-dual partial functions on 2, and the problem of describing them all wasposed by some authors.In this paper we show that the set of partial clones that contain allmonotone self-dual partial functions is of continuum cardinality on 2.1. PreliminariesLet A be a finite non-singleton set. Without loss of generality we assumethat A = k := {0, . . . , k − 1}. For a positive integer n, an n-ary partialfunction on k is a map f : dom (f) → k where dom (f) is a subset of kncalled the domain of f . Let Par(n)(k) denote the set of all n-ary partialfunctions on k and letPar(k) :=∪n≥1Par(n)(k).This work was done while the first named author was a Research Assistant at the Math-ematics Research Unit at the University of Luxembourg, Luxembourg, and an AssociateProfessor at LAMSADE at Université Paris-Dauphine, France.The second named author wishes to acknowledge the financial support by ARP grants.Les trois auteurs veulent rendre hommage à Maurice Pouzet. Maurice est un mathé-maticien brillant et respecté, ses contributions scientifiques sont immenses. Tous ceux quile connaissent apprécient non seulement sa profonde culture scientifique mais aussi sesgrandes qualités humaines. Maurice est plein dǵeards pour son entourage, il est généreuxet très d évoué envers ses amis. Il prend plaisir à encourager, aider et rendre service auxautres. Nous avons la chance et le privilège davoir Maurice comme collaborateur et ami.Merci pour tout Maurice!12 M. COUCEIRO, L. HADDAD, AND I. G. ROSENBERGFor n,m ≥ 1, f ∈ Par(n)(k) and g1, . . . , gn ∈ Par(m)(k), the compositionof f and g1, . . . , gn, denoted by f [g1, . . . , gn] ∈ Par(m)(k), is defined bydom (f [g1, . . . , gn]) := {a⃗ ∈ km | a⃗ ∈m∩i=1dom (gi)and (g1(⃗a), . . . , gm(⃗a)) ∈ dom (f)}andf [g1, . . . , gn](⃗a) := f(g1(⃗a), . . . , gn(⃗a)),for all a⃗ ∈ dom (f [g1, . . . , gn]).For every positive integer n and each 1 ≤ i ≤ n, let eni denote the n-aryi-th projection function defined byeni (a1, . . . , an) = aifor all (a1, . . . , an) ∈ kn. Furthermore, letJk := {eni : 1 ≤ i ≤ n}be the set of all (total) projections.Definition 1. A partial clone on k is a composition closed subset of Par(k)containing Jk.Remark 1. There are two other equivalent definitions for partial clones.One definition uses Mal’tsev’s formalism and the other uses the concept ofone point extension. These definitions can be found in chapter 20 of [7].The partial clones on k, ordered by inclusion, form a lattice LPk in whichthe infinimum is the set-theoretical intersection. That means that the inter-section of an arbitrary family of partial clones on k is also a partial clone onk. A maximal partial clone on k is a coatom of the lattice LPk . Thereforea partial clone M is maximal if there is no partial clone C over k such thatM ⊂ C ⊂ Par(k).Example 1. The set of partial functionsΩk :=∪n≥1{f ∈ Par(n)(k) | dom (f) ̸= ∅ =⇒ dom (f) = kn}is a maximal partial clone on k.Definition 2. For h ≥ 1, let ρ be an h-ary relation on k and f be an n-ary partial function on k. We say that f preserves ρ if for every h × nmatrix M = [Mij ] whose columns M∗j ∈ ρ, (j = 1, . . . n) and whose rowsMi∗ ∈ dom (f) (i = 1, . . . , h), the h-tuple (f(M1∗), . . . , f(Mh∗)) ∈ ρ. DefinepPol ρ := {f ∈ Par(k) | f preserves ρ}.It is well known that pPol ρ is a partial clone called the partial clone deter-mined by the relation ρ.PARTIAL CLONES CONTAINING BOOLEAN MONOTONE SELF-DUAL P. FUNCT.S 3Notice that if there is no h×n matrix M = [Mij ] whose columns M∗j ∈ ρand whose rows Mi∗ ∈ dom (f), then f ∈ pPol ρ.Example 2. Let 2 := {0, 1} and let {(0, 0), (0, 1), (1, 1)} be the natural orderon 2. Consider the binary relation {(0, 1), (1, 0)} on 2. ThenpPol {(0, 0), (0, 1), (1, 1)}is the set of all monotone partial functions andpPol {(0, 1), (1, 0)}is the set of all self-dual partial functions on 2.For simplicity we will write pPol (≤) and pPol ( ̸=) forpPol ({(0, 0), (0, 1), (1, 1)}) and pPol ({(0, 1), (1, 0)}),respectively. It is not difficult to see thatpPol (≤) := {f ∈ Par(2) | [a,b ∈ dom (f),a ≤ b] =⇒ f(a) ≤ f(b)}, andpPol ( ̸=) := {f ∈ Par(2) | [a,a + 1 ∈ dom (f)] =⇒ f(a + 1) = f(a) + 1}where the above sums are taken mod 2.As mentioned earlier, Freivald showed that there are exactly eight max-imal partial clones on 2. The following two relations are needed to stateFreivald’s result. SetR1 := {(x, x, y, y) | x, y ∈ 2} ∪ {(x, y, y, x) | x, y ∈ 2} andR2 := R1 ∪ {(x, y, x, y) | x, y ∈ 2}.Theorem 2 ([2]). There are exactly 8 maximal partial clones on 2, namely,pPol {0}, pPol {1}, pPol {(0, 1)}), pPol (≤), pPol ( ̸=), pPol (R1), pPol (R2),and Ω2.Notice that the total functions in pPolR2 (i.e., the functions with fulldomain) form the maximal clone of all (total) linear functions over 2 (see,e.g., chapter 3 of [7]).An interesting and somehow difficult problem in clone theory is to studyintersections of maximal partial clones. It is shown in [1] that the set ofall partial clones on 2 that contain the maximal clone consisting of all totallinear functions on 2 is of continuum cardinality (for details see [1, 4] andTheorem 20.7.13 of [7]). A consequence of this is that the interval of partialclones [pPol (R2) ∩ Ω2,Par(2)] is of continuum cardinality on 2.A similar result, (but slightly easier to prove) is established in [3] whereit is shown that the interval of partial clones [pPol (R1) ∩ Ω2,Par(2)] isalso of continuum cardinality. Notice that the three maximal partial clonespPolR1, pPolR2 and Ω2 contain all unary functions (i.e., maps) on 2. Suchpartial clones are called S lupecki type partial clones in [4, 10]. These are theonly three maximal partial clones of S lupecki type on 2.For a complete study of the pairwise intersections of all maximal partialclones of S lupecki type on a finite non-singleton set k, see [4].4 M. COUCEIRO, L. HADDAD, AND I. G. ROSENBERGThe papers [5, 6, 8, 12, 13] focus on the case k = 2 where various inter-esting, and sometimes hard to obtain, results are established.For instance, the intervals[pPol {0} ∩ pPol {1} ∩ pPol {(0, 1)} ∩ pPol (≤), Par(2)] and[pPol {0} ∩ pPol {1} ∩ pPol {(0, 1)} ∩ pPol ( ̸=), Par(2)]are shown to be finite and are completely described in [5]. Some of theresults in [5] are included in [12, 13] where partial clones on 2 are handledvia the one point extension approach (see Section 20.2 in [7]).In view of results from [1, 3, 5, 12, 13], it was thought that if 2 ≤ i ≤ 5 andM1, . . . ,Mi are non-S lupecki maximal partial clones on 2, then the interval[M1 ∩ · · · ∩Mi,Par(2)]is either finite or countably infinite.Now it was shown in [6] that the interval of partial clones [pPol (≤) ∩pPol ( ̸=),Par(2)] is infinite. This result is mentioned in Theorem 20.8 of[7] (with an independent proof given in [8]) and in chapter 8 of the PhDthesis [11]. However, it remained an open problem to determine whether[pPol (≤) ∩ pPol ( ̸=),Par(2)] is countably or uncountably infinite.In this paper we settle this question by proving that the interval of partialclones[pPol (≤) ∩ pPol ( ̸=),Par(2)]is of continuum cardinality on 2.2. The constructionFor n ≥ 5 and n > k > 1 we denote by σnk ⊆ 22n the (2n)-ary relationdefined byσnk := {(x1, . . . , xn, y1, . . . , yn) ∈ 22n | ∀ i = 1, . . . , n, xi ̸= yi, and∀ i = 1, . . . , n, yi+1 ≤ xi and yi+2 ≤ xi . . . and yi+k ≤ xi},where the subscripts i + j in the above definition are taken modulo n. It isnot difficult to see thatσnk := {(x1, . . . , xn, y1, . . . , yn) ∈ 22n | ∀ i = 1, . . . , n, xi ̸= yi, and∀ i = 1, . . . , n, xi = 0 =⇒ [xi+1 = xi+2 = · · · = xi+k = 1]}.By the Definability Lemma established by B. Romov in [9] (see also Lemma20.3.4 in [7] and [4, 5, 6] for details), we have thatpPol (≤) ∩ pPol ( ̸=) ⊆ pPol (σnk )for all n ≥ 5 and all k ≥ 1.For n ≥ 5 and n > k ≥ 1, we denote by ρnk ⊆ 24n the (4n)-ary relationdefined byρnk := {(x1, . . . , xn, xn+1, . . . , x2n, y1, . . . , yn, yn+1, . . . , y2n) ∈ 24n |(x1, . . . , xn, y1, . . . , yn) ∈ σn1 , and (xn+1, . . . , x2n, yn+1, . . . , y2n) ∈ σnk}.PARTIAL CLONES CONTAINING BOOLEAN MONOTONE SELF-DUAL P. FUNCT.S 5Again by the Definability Lemma, we have thatpPol (σn1 ) ∩ pPol (σnk ) ⊆ pPol (ρnk),and thus pPol (≤) ∩ pPol ( ̸=) ⊆ pPol (ρnk) for all n ≥ 5 and all k ≥ 1.Our goal is to construct an infinite set of odd integers X and an infinitefamily of partial functions {gt, t ∈ X} so that for every t, t′ ∈ X, we havegt ∈ pPol ρn(t′)t′ if and only if t ̸= t′.Remark 3. Since every tuple in σnk (resp. ρnk) is completely determined byits first n entries (resp. 2n entries), we will omit the second half of suchtuples. We therefore denote by Snk and Rnk the relations obtained from σnkand ρnk , respectively, by deleting the second half of every tuple in σnk and ρnk ,i.e.,Snk := {(x1, . . . , xn) ∈ 2n | (x1, . . . , xn, 1 + x1, . . . , 1 + xn) ∈ σnk}andRnk := {(x1, . . . , x2n) ∈ 22n | (x1, . . . , x2n, 1 + x1, . . . , 1 + x2n) ∈ ρnk}where the above sums are taken mod 2.Note that Snk is the n-ary relation on 2 whose members are tuples in whichany two 0’s are separated by at least k symbols 1 (in particular, if the firstposition is 0, then the last k positions must be 1). Furthermore, Rnk is thecartesian product Sn1 × Snk .As mentioned earlier we will use the relations Sn1 , Snk and Rnk with theunderstanding that we are omitting the second parts of the relations σn1 , σnkand ρnk in order to simplify the notation.Notations. In the sequel k ≥ 4 stands for an even integer. Set n(k) :=k(k + 1) + 1. We will write ρn(k) for ρn(k)k and Rn(k) for Rn(k)k . Let Mk↑be the n(k) × n(k) matrix with columns in Sn(k)1 , the first being c1 =[0110101 . . . 0101]T and the remaining columns are obtained by applyingcyclic shifts to c1, i.e.,c2 = [10110101 · · · 010]T ,c3 = [010110101 · · · 01]T ,· · ·cn(k) = [110101 · · · 010]T .Remark 4. Let ri and rj be two rows of Mk↑ . If |i− j| ≥ 2(modn(k)), thenri and rj have a 0 in the same position.Lemma 5. If k′ < k, then there is no n(k′)×n(k) matrix N whose columnsare in Sn(k′)1 and whose rows are rows of Mk↑ .Proof. Suppose that k′ < k and that N is an n(k′)×n(k) matrix whosecolumns are in Sn(k′)1 . Suppose, by way of contradiction, that the rows of N6 M. COUCEIRO, L. HADDAD, AND I. G. ROSENBERGare rows of Mk↑ . By Remark 4, the only possible adjacent rows of a row rin N are exactly the predecessor and successor rows of r in Mk↑ . But thenn(k′) would be even, thus yielding the desired contradiction.Let Mk↓ be the n(k) × n(k) matrix with columns in Sn(k)k , and such thatthe first is c′1 = [0 1 · · · 1︸ ︷︷ ︸k+10 1 · · · 1︸ ︷︷ ︸k· · · 0 1 · · · 1︸ ︷︷ ︸k]T and the remaining columns areobtained by applying cyclic shifts to c′1 as before.Remark 6. Since k ≥ 4 is even, if ri is a row of Mk↑ , and r′j is a row ofMk↓ , then ri and r′j have a 0 in the same position.Lemma 7. If k′ > k, then there is no n(k′)×n(k) matrix N whose columnsare in Sn(k′)k and whose rows are rows of Mk↓ .Proof. Suppose that k′ > k and that N is an n(k′)×n(k) matrix whosecolumns are in Sn(k′)k . Assume, by way of contradiction, that the rows of Nare rows of Mk↓ . Since each row of Mk↓ has exactly k 0’s, we have that N hask×n(k′) 0’s. Hence the matrix N has a column with at least k×n(k′)n(k) symbols0. It is easy to verify that since k′ > k ≥ 4, we have that k×n(k′)n(k) > k′. Butthis yields the desired contradiction, since all columns of N are members ofSn(k′)k , and each has at most k′ 0’s.Define Mk as the 2n(k) × n(k) matrix given byMk =(Mk↑Mk↓).Notice that each column of Mk is a tuple of Rn(k).Lemma 8. Let N be a 2n(k′) × n(k) matrix whose columns are in Rn(k′)and whose rows are rows of Mk. Then, either all rows of N are rows of Mk↓ ,or the first n(k′) are rows of Mk↑ and the remaining n(k′) are rows of Mk↓ .Proof. By Remark 4 and the fact that Rn(k′) := Sk′1 ×Sk′k , there cannotbe more than 2 rows of Mk↑ among the last n(k′). In fact, by Remark 6 therecan only be rows from Mk↓ among the last n(k′) rows of N . Furthermore,from Remark 6 and the fact that Rn(k′) := Sk′1 × Sk′k , it follows that eitherall of the first n(k′) rows of N are rows of Mk↑ or all of the first n(k′) rowsof N are rows of Mk↓ .Let fk be the n(k)-ary partial function whose domain is the set of rowsof Mk, and such that fk is constant 0 on the rows of Mk↑ and constant 1 onthe rows of Mk↓ .Theorem 9. Let k, k′ ≥ 4 be even integers. Then fk ∈ pPolRn(k′) if andonly if k ̸= k′.PARTIAL CLONES CONTAINING BOOLEAN MONOTONE SELF-DUAL P. FUNCT.S 7Proof. Since [0 · · · 01 · · · 1]T does not belong to Rn(k), we see that fk ̸∈pPolRn(k).So suppose that k ̸= k′. If k < k′, then it follows from Definition 2 andLemmas 7 and 8 that fk ∈ pPolRn(k′).Suppose now that k > k′. If N is an 2n(k′)×n(k) matrix whose columnsare in Rn(k′) and whose rows are rows of Mk (otherwise we are done for thedomain of fk is exactly the set of rows of Mk), then by Lemmas 5 and 8 itfollows that all rows of N are rows of Mk↓ . Since fk is constant 1 on the rowsof Mk↓ , and since the constant 1 2n(k′) tuple belongs to Rn(k′), we concludethat fk ∈ pPolRn(k′).Let Mk be the 2n(k) × n(k) matrix obtained by replacing every row ofthe matrix Mk by its dual tuple (obtained by interchanging 1’s and 0’s) anddefine Lk as the 4n(k) × n(k) matrix given byLk =(MkMk).Moreover, let gk be the n(k)-ary partial function whose domain is theset of rows of Lk, and such that gk(u⃗) = fk(u⃗) if u⃗ is a row of Mk andgk(u⃗) = 1 + fk(u⃗) (mod 2) if u⃗ is a row of Mk. Then, Theorem 9 can berestated as follows:Main Theorem. Let k, k′ ≥ 4 be even integers. Then gk ∈ pPol ρn(k′) ifand only if k ̸= k′.Let E≥4 := {4, 6, 8, . . . } be the set of all even integers greater or equal to4 and denote by P(E≥4) the power set of E≥4. SincepPol (≤) ∩ pPol ( ̸=) ⊆ pPol (ρnk)for every n ≥ 5 and every n > k ≥ 1, we havepPol (≤) ∩ pPol ( ̸=) ⊆∩t∈E≥4\XpPol ρn(t)for every subset X of E≥4.So let X ⊂ E≥4 and fix k ∈ X. Then gk ∈ pPol ρn(t) for all t ∈ E≥4 \X,i.e.,gk ∈∩t∈E≥4\X.On the other hand, if k ∈ E≥4 \X, then we havegk ̸∈∩t∈E≥4\XpPol ρn(t) and gk ̸∈∩t ̸∈XpPol ρn(t),since gk ̸∈ pPol ρn(k). Therefore the mapχ := P(E≥4) → [pPol (≤) ∩ pPol ( ̸=),Par(2)]8 M. COUCEIRO, L. HADDAD, AND I. G. ROSENBERGdefined byχ(X) :=∩t∈E≥4\XpPol ρn(t)is one-to-one and we have shown the following result which answers ourquestion on cardinality of the interval [pPol (≤) ∩ pPol ( ̸=),Par(2)].Corollary 10. The interval of partial clones [pPol (≤) ∩ pPol ( ̸=),Par(2)]is of continuum cardinality on 2.References[1] V. B. Alekzeev and L. L. Voronenko, Some closed classes in the partial two-valuedlogic (Russian), Diskret. Mathematika 6, 4 (1994), 58–79[2] R. V. Freivald, Completness criteria for functions of the algebra of logic and many-valued logics. Dokl. Akad. Nauk. SSSR, 167, 6 (1966) 1249-1250.[3] L . Haddad, On the depth of the intersection of two maximal partial clones, Multi.Val. Logic, an International Journal, 3, No 3 (1998), 259–270.[4] L. Haddad and D. Lau, Pairwise intersections of S lupecki type maximal partial clones.Contributions to Algebra and Geometry, Volume 41 (2000), No. 2, 537 – 555.[5] L. Haddad and G. E. Simons, Interval of Boolean Partial Clones. Italian Journal ofPure and App. Math. , No 21 (2007), pp 147-162.[6] L. Haddad, Partial clones containing all self-dual monotonic Boolean partial functions.Proc. 39th IEEE International Symposium on Multiple-Valued Logic, pp 173 – 178, Ok-inawa, Japan, May 2009. Detailled paper to appear in Journal of Multiple-Valued Logicand Soft Computing.[7] D. Lau, Function Algebras on Finite Sets, a basic course on Multiple-Valued Logic andClone Theory, 670 pages, Springer Monograph in Mathematics, 2006.[8] D. Lau and K. Schölzel, A Classification of Partial Boolean Clones. Proc. 40th IEEEInternational Symposium on Multiple-Valued Logic, pp 198 – 194, Barcelona, Spain, May2010.[9] B. A. Romov, The algebras of partial functions and their invariants, Kibernetika;English translation in Cybernetics 17 (1981) 157–167.[10] B. A. Romov, Maximal subalgebras of algebras of partial multivalued logic functions,Kibernatika; English translation in Cybernetics 16 (1980) 31-41.[11] K. Schölzel, Clones of Partial Functions on Finite Sets, PhD Thesis. University ofRostock. Published as Karsten Schölzel - Clones of Partial Functions on Finite Sets(ISBN: 978-3-8322-9823-4) by Shaker Verlag.[12] B. Strauch, On partial classes containing all monotone and zero-preserving totalBoolean functions. Math. Log. Quart. 43 (1997).[13] B. Strauch, The classes which contain all monotone and idempotent total Booleanfunctions, Universität Rostock, preprint 1996.PARTIAL CLONES CONTAINING BOOLEAN MONOTONE SELF-DUAL P. FUNCT.S 9(Miguel Couceiro) LORIA (CNRS - Inria Nancy G.E. - Université de Lor-raine), Equipe Orpailleur, Batiment B, Campus Scientifique, B.P. 239, F-54506Vandoeuvre-lès-NancyE-mail address: miguel.couceiro@inria.fr(Lucien Haddad) Départment de Mathématiques et dInformatique, Collègemilitaire royal du Canada, boite postale 17000, STN Forces, Kingston ONK7K 7B4 Canada.E-mail address: haddad-l@rmc.ca(Ivo G. Rosenberg) Départment de Mathématiques et Statistique, Universitéde Montrál, Montréal, Québec, H3C 3J7 CanadaE-mail address: rosenb@dms.umontreal.ca

Partial clones containing all Boolean monotone self-dual partial functions

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Partial clones containing all Boolean monotone self-dual partial functions

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