Pattern Structures and Their Morphisms

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and Their Morphisms

Lars Lumpe¹and Stefan E. Schmidt² Institut f¨ur Algebra,

Technische Universit¨at Dresden [email protected]¹, [email protected]²

Abstract. Projections of pattern structures don’t always lead to pattern structures, however residual projections and o-projections do. As a unifying approach, we introduce the notion of pattern morphisms between pattern structures and pro- vide a general sufficient condition for a homomorphic image of a pattern structure being again a pattern structure. In particular, we receive a better understanding of the theory of o-projections.

1 Introduction

Pattern structures within the framework of formal concept analysis have been introduced in [3]. Since then they have turned out to be a useful tool for analysing various real-world applications (cf. [3–7]). In this work we want to point out that the theoretical foundations of pattern structures encourage still some fruitful discussions. In particular, the role projections play within pattern structures for information reduction still needs some further investigation.

The goal of our paper is to establish an adequate concept of pattern morphism between pattern structures, which also gives a better understanding of the concept of o- projections as recently introduced and investigated in [2]. In [8], we showed that projections of pattern structures do not necessarily lead to pattern structures again, however, residual projections do. It turns out that the concept of residual maps between the posets of patterns (w.r.t. two pattern structures) gives the key for a unifying view of o-projections and residual projections.

We also derive that a pattern morphism from a pattern structure to a pattern setup (introduced in this paper), which is surjective on the sets of objects, yields again a pattern structure.

Our main result states that a pattern morphism always induces an adjunction between the corresponding concept lattices. In case the underlying map between the sets of objects is surjective, the induced residuated map between the concept lattices turns out to be surjective too.

The fundamental order theoretic concepts of our paper are nicely presented in the book onResiduation Theoryby T.S. Blythe and M.F. Janowitz (cf. [1]).

c paper author(s), 2015. Published in Sadok Ben Yahia, Jan Konecny (Eds.): CLA 2015, pp. 171–179, ISBN 978–2–9544948–0–7, Blaise Pascal University, LIMOS laboratory, Clermont-Ferrand, 2015. Copying permitted only for private and academic purposes.

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2 Preliminaries

Definition 1(Adjunction). LetP pP,¤qandL pL,¤q be posets; furthermore let f :PÑL and g:LÑP be maps.

(1) The pairpf,gqis anadjunctionw.r.t.pP,Lqif f x¤y is equivalent to x¤gy for all xPP and yPL. In this case, we will refer topP,L,f,gqas aposet adjunction.

(2) f isresiduatedfromPtoLif the preimage of a principal ideal inLunder f is always a principal ideal inP, that is, for every yPL there exists xPP s.t.

f¹ttPL|t¤yu tsPP|s¤xu.

(3) g isresidualfromLtoPif the preimage of a principal filter inPunder g is always a principal filter inL, that is, for every xPP there exists yPL s.t.

g¹tsPP|x¤su ttPL|y¤tu.

(4) The dual ofLis given byL^op pL,¥qwith ¥: tpx,tq PLL|t¤xu. The pair pf,gqis aGalois connectionw.r.t.pP,Lqifpf,gqis anadjunctionw.r.t.pP,L^opq.

The following well-known facts are straightforward (cf. [1]).

Proposition 1. LetP pP,¤qandL pL,¤qbe posets.

(1) A map f:PÑL is residuated fromPtoLiff there exists a map g:LÑP s.t.pf,gq is an adjunction w.r.t.pP,Lq.

(2) A map g:LÑP is residual fromLtoPiff there exists a map f :PÑL s.t.pf,gq is an adjunction w.r.t.pP,Lq.

(3) Ifpf,gqandph,kqare adjunctions w.r.t.pP,Lqwith f h or gk then fh and gk.

(4) If f is a residuated map fromPtoL, then there exists a unique residual map f fromLtoPs.t.pf,f qis an adjunction w.r.t.pP,Lq. In this case, f is called the residual mapof f .

(5) If g is a residual map fromLtoP, then there exists a unique residuated map g fromPtoLs.t.pg,gqis an adjunction w.r.t.pP,Lq. In this case, gis called the residuated mapof g.

Definition 2. LetP pP,¤qbe a poset and TP. Then

(1) TherestrictionofPonto T is given byP|T: pT,¤ XpTTqq, which clearly is a poset too.

(2) Thecanonical embeddingofP|T intoPis given by the map T ÑP,tÞÑt.

(3) T is akernel systeminPif the canonical embeddingτofP|T intoPis residuated.

In this case, the residual mapϕofτwill also be called theresidual mapof T inP. The compositionκ:τϕis referred to as thekernel operatorassociated with T inP.

(4) Dually, T is aclosure system in P if the cannonical embedding τ of P|T into P is residual. In this case, the residuated map ψ of τ will also be called the residuated mapof T inP. The compositionγ:τψis referred to as theclosure operatorassociated with T inP.

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(5) A mapκ:PÑP is akernel operatoronPif s¤x is equivalent to s¤κx for all sPκP and xPP.

Remark: In this case,κP forms a kernel system inP, the kernel operator of which isκ.

(6) Dually, a mapγ:PÑP is aclosure operatoronPif x¤t is equivalent toγx¤t for all xPP and tPγP.

Remark: In this case,ϕP forms a closure system inP, the closure operator of which isγ.

The following known facts will be needed for the sequel (cf. [1]) . Proposition 2. LetP pP,¤qandL pL,¤qbe posets.

(1) If f is a residuated map fromPtoLthen f preserves all existing suprema inP, that is, if sPP is the supremum (least upper bound) of XP inPthen f s is the supremum of f X inL.

In casePandLare complete lattices, the reverse holds too: If a map f fromPto Lpreserves all suprema, that is,

fpsup_PXq sup_L f X f or all XP, then f is residuated.

(2) If g is a residual map fromLtoP, then g preserves all existing infima inL, that is, if tPL is the infimum (greatest lower bound) of YL inLthen gt is the infimum of gY inP.

In casePandLare complete lattices, the reverse holds too: If a map g fromLtoP preserves all infima, that is,

fpinf_PYq inf_LgY f or all Y L, then g is residual.

(3) For an adjunctionpf,gqw.r.t.pP,Lqthe following hold:

(a1) f is an isotone map fromPtoL. (a2) fgf f

(a3) f P is a kernel system inLwith fg as associated kernel operator onL. In particular, LÑP,yÞÑ f gy is a residual map fromLtoL|f P.

(b1) g is an isotone map fromLtoP. (b2) gfgg

(b3) gL is a closure system inPwith gf as associated closure operator onP. In particular, PÑgL,xÞÑg f x is a residuated map fromPtoP|gL.

3 Adjunctions and Their Concept Posets

Definition 3. LetP: pP,S,σ,σ qandQ : pQ,T,τ,τ qbe poset adjunctions. Then a pairpα,βqforms a morphism fromPtoQifpP,Q,α,α qandpS,T,β,β qare poset adjunctions satisfying

ταβσ

Remark: This impliesα τ σ β , that is, the following diagrams are commutative:

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P Q

T S

α

β

τ σ

P Q

T S

α

β

τ σ

Next we illustrate the involved poset adjunctions:

P Q

T S

α α

β β

τ τ

σ σ

Definition 4(Concept Poset). For a poset adjunctionP pP,S,σ,σ qlet BP: tpp,sq PPS|σps^σ spu

denote the set ofpformalqconceptsinP. Then theconcept posetofP is given by BP: pPSq |BP,

that is,pp0,s0q ¤ pp1,s1qholds iff p0¤p1iff s0¤s1, for allpp0,s0q,pp1,s1q PBP. If pp,sqis a formal concept inPthen p is referred to asextentinPand s asintentinP. Theorem 1. Letpα,βqbe a morphism from a poset adjunctionP pP,S,σ,σ qto a poset adjunctionQ pQ,T,τ,τ q. Then

pBP,BQ,Φ,Ψq is a poset adjunction for

Φ: BPÑBQ,pp,sq ÞÑ pτ βs,βsq and Ψ: BQ ÑBP,pq,tq ÞÑ pα q,σα qq.

In addition, ifαis surjective then so isΦ.

Remark: In particular we want to point out thatα q is an extent inP for every extent q inQ and similarly,βs is an intent inQ for every intent s inP.

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Proof. Letpp,sq PBP and pq,tq PBQ; thenσps and σ sp and τqt and τ tq. This impliesβsβσpταp, thus

Φpp,sq pτ βs,βsq PBP (sinceττ βsττ ταpταpβsq. Similarly,Ψpq,tq PBQ.

Assume now thatΦpp,sq ¤ pq,tqholds, which impliesβs¤t. It follows that ταpβσpβs¤t

and hence

p¤α τ tα q, that is,pp,sq ¤Ψpq,tq.

Conversely, assume thatpp,sq ¤Ψpq,tqholds, which impliesp¤α q. It follows that p¤α qα τ tσ β t,

and henceβsβσp¤t, that is,Φpp,sq ¤ pq,tq.

Assume now thatαis surjective; thenαα idQ. Letpq,tq PBP, that is,τqtand τ tq. Then for p:α qands:σpwe havepp,sq PBPsince

σ sσ σα qσ σα τ tσ σσ β tσ β tα τ tα qp.

Our claim is now thatΦpp,sq pq,tqholds, that is,βst. The latter is true, since αpαα qqimplies

βsβσpταpτqt.

Discussion for clarification: The question was raised whether, in the previous theorem, the residuated mapΦfrom BPto BQ allows some modification, since the map

PSÑQT,pp,sq ÞÑ pαp,βsq

is obviously residuated fromPStoQT. However, in general the latter map does not restrict to a map from BP to BQ. Indeed, our construction of the mapΦ is of the formpp,sq ÞÑ pα¹p,βsq. As a warning, we want to point out that, in general, there is no residuated map from BP to BQ of the formpp,sq ÞÑ pαp,β¹sq. The simple reason for this is thatβsis an intent inQ for every intentsinP, while there may exist an extentp inP such thatαpis not an extent inQ.

4 Morphisms between Pattern Structures

Definition 5. A tripleG pG,D,δqis apattern setupif G is a set,D pD,qis a poset, andδ :GÑD is a map. In case every subset ofδG: tδg | gPGuhas an infimum inD, we will refer toGaspattern structure. Then the set

CG: tinf_DδX |XGu

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forms a closure system inD.

IfG pG,D,δqandH pH,E,εqeach is a pattern setup, then a pairpf,ϕqforms a pattern morphismfromGtoH if f:GÑH is a map andϕis a residual map fromD toEsatisfyingϕδ εf , that is, the following diagram is commutative:

G H

E D

f

ϕ δ ε

In the sequel we show how our previous considerations apply to pattern structures.

Applications

(1) LetGbe a pattern structure andH be a pattern setup. Ifpf,ϕqis a pattern morphism fromG toH with f being surjective, thenH is also a pattern structure.

(2) LetG pG,D,δq andH pH,E,εqbe pattern structures. Also let pf,ϕq be a pattern morphism fromGtoH.

To apply the previous theorem we give the following construction:

f gives rise to an adjunctionpα,α qbetween the power set lattices2^G: p2^G,q and2^H: p2^H,qvia

α: 2^GÑ2^H,XÞÑ f X

and α : 2^HÑ2^G,YÞÑ f¹Y.

Further letϕdenote the residuated map ofϕw.r.t.pE,Dq, that is,pE,D,ϕ,ϕqis a poset adjunction. Then, obviously,pD^op,E^op,ϕ,ϕqis a poset adjunction too.

For pattern structures the following operators are essential:

: 2^GÑD,XÞÑinf_DδX

:DÑ2^G,dÞÑ tgPG|dδgu

: 2^HÑE,ZÞÑinf_EεZ

:EÑ2^H,eÞÑ thPH|eεhu

It now follows thatpα,ϕqforms a morphism from the poset adjunction P p2^G,D^op,,q

to the poset adjunction

Q p2^H,E^op,,q.

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In particular,pf XqϕpXqholds for allXG.

Here we give an illustration of the constructed adjunctions:

2

^G

2

^H

E

^op

D

^op

α α

ϕ ϕ

ReplacingD^opbyDandE^opbyEwe receive the following commutative diagrams:

2^G 2^H

E D

α

ϕ

2^G 2^H

E D

α

ϕ

In combination we receive the following diagram of Galois connections and adjunctions between them:

2

^G

2

^H

E D

α α

ϕ ϕ

l

For the following we recollect that theconcept latticeofGis given byBG:BP

— similarly,BH :BQ.

Now we are prepared to give an application of Theorem 1 to concept lattices of pattern structures:pBG,BH,Φ,Ψqis an adjunction for

Φ: BGÑBH,pX,dq ÞÑ ppϕdq,ϕdq

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and

Ψ: BH ÑBG,pZ,eq ÞÑ pf¹Z,pf¹Zqq.

In case f is surjective,Φis surjective too.

Remark: This application implies a generalization of Proposition 1 in [2], that is, if Zis an extent inH, then f¹Zis an extent inG, and ifdis an intent inG thenϕd is an intent inH.

(3) LetG pG,D,δqbe a pattern structure and letκbe a kernel operator onD. Then ϕ:DÑκD,dÞÑκdforms a residual map fromDtoκD:D|κD, andpidG,ϕq is a pattern morphism fromGtoH : pG,κD,ϕδq.

Remark: In [2], ϕ is called an o-projection. The above clarifies the role of o- projections for pattern structures.

(4) LetG pG,D,δqbe a pattern structure, and letκ be a residual kernel operator on D. ThenpidG,κqis a pattern morphism fromGtoH : pG,D,κδq.

Remark: In [8],κis also referred to as a residual projection. The above clarifies the role of residual projections for pattern structures.

(5) Generalizing [2] and [8], we observe that ifG pG,D,δqis a pattern structure and ϕ is a residual map fromDtoE, thenpidG,ϕqis a pattern morphism fromG to H pG,E,ϕδqsatisfying that

Φ: BGÑBH,pX,dq ÞÑ ppϕdq,ϕdq is a surjective residuated map fromBG toBH.

In particular,XϕpX) holds for allXG.

Remark: This application gives a better understanding to properly generalize the concept of projections as discussed in [3] and subsequently in [2, 4–8].

References

1. T.S. Blyth, M.F.Janowitz (1972), Residuation Theory, Pergamon Press, pp. 1-382.

2. A. Buzmakov, S. O. Kuznetsov, A. Napoli (2015) , Revisiting Pattern Structure Projections.

Formal Concept Analysis. Lecture Notes in Artificial Intelligence (Springer), Vol. 9113, pp 200-215.

3. B. Ganter, S. O. Kuznetsov (2001), Pattern Structures and Their Projections. Proc. 9th Int.

Conf. on Conceptual Structures, ICCS01, G. Stumme and H. Delugach (Eds.). Lecture Notes in Artificial Intelligence (Springer), Vol. 2120, pp. 129-142.

4. T. B. Kaiser, S. E. Schmidt (2011), Some remarks on the relation between annotated ordered sets and pattern structures. Pattern Recognition and Machine Intelligence. Lecture Notes in Computer Science (Springer), Vol. 6744, pp 43-48.

5. M. Kaytoue, S. O. Kuznetsov, A. Napoli, S. Duplessis (2011), Mining gene expression data with pattern structures in formal concept analysis. Information Sciences (Elsevier), Vol.181, pp. 1989-2001.

6. S. O. Kuznetsov (2009), Pattern structures for analyzing complex data. In H. Sakai et al.

(Eds.). Proceedings of the 12th international conference on rough sets, fuzzy sets, data mining and granular computing (RSFDGrC09). Lecture Notes in Artificial Intelligence (Springer), Vol. 5908, pp. 33-44.

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7. S. O. Kuznetsov (2013), Scalable Knowledge Discovery in Complex Data with Pattern Structures. In: P. Maji, A. Ghosh, M.N. Murty, K. Ghosh, S.K. Pal, (Eds.). Proc. 5th Inter- national Conference Pattern Recognition and Machine Intelligence (PReMI2013). Lecture Notes in Computer Science (Springer), Vol. 8251, pp. 30-41.

8. L. Lumpe, S. E. Schmidt (2015), A Note on Pattern Structures and Their Projections. For- mal Concept Analysis. Lecture Notes in Artificial Intelligence (Springer), Vol. 9113, pp 145-150.