Morphisms Between Pattern Structures and Their Impact on Concept Lattices

Morphisms Between Pattern Structures and Their Impact on Concept Lattices

Lars Lumpe¹and Stefan E. Schmidt²

Institut f¨ur Algebra, Technische Universit¨at Dresden larslumpe@gmail.com¹, midt1@msn.com²

Abstract. We provide a general framework for pattern structures by investigat- ing adjunctions between posets and their morphisms. Our special interest is the impact of pattern morphisms on the induced concept lattices. In particular we are interested in conditions which are sufficient for the induced residuated maps to be injective, surjective or bijective. One application is that every representation context of a pattern structure has a formal concept lattice that is induced by a certain pattern morphism.

1 Introduction

Pattern structures within the framework of formal concept analysis have been intro- duced in [3]. Since then they have turned out to be a useful tool for analysing various real-world applications (cf. [3–7]). Our paper extends the concept of representation contexts and interprets them via morphisms, closely related to o-projections as recently introduced and investigated in [2]. In [8], we disussed the meaning of projections of pattern structures, realizing the importance of residual projections. As a matter of fact, our generalization of representation contexts of pattern structures gives rise to residual projections.

2 Preliminaries

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]).

Definition 1(Adjunction). LetP pP,¤qandL pL,¤qbe 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.

(6) A residuated map f fromPtoLis surjective iff ff id_Liff f is injective.

(7) A residuated map f fromPtoLis injective iff ff id_Liff f is surjective.

Definition 2. LetP pP,¤qbe a poset and T„P. 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 TÑP,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 a closure 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.

(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 X„P 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 X„P, 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 Y„L 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 commu- tative:

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.

From [9] we point out Theorem 1:

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,Φ q

is a poset adjunction for

Φ: BPÑBQ,pp,sq ÞÑ pτ βs,βsq and

Φ : BQ ÑBP,pq,tq ÞÑ pα q,σ α qq.

P Q

T S

α

β

τ

σ BP Φ BQ

Theorem 2. Under the conditions of the previous theorem the following hold:

(1) Ifα is surjective thenΦis surjective too.

(2) Ifβ is injective thenΦis injective too.

(3) Ifα is surjective andβis injective thenΦis an isomorphism fromBPtoBQ..

Proof. (1) Assume thatα is surjective, that is,αα id_Q. Then for allpp,sq PBP, the second component ofpΦΦ qpp,sqisβ σ α qτ α α qτqs. This yields ΦΦ id_BQ, that is,Φ is surjective.

(2) The first component ofpΦ Φqpp,sqisα τ βsτ β βsτ sp. There- fore,Φ Φid_BP, which yieldsΦ being injective.

(3) Ifαis surjective andβis injective, thanΦandΦ are naturally inverse by (1) and (2), that is,Φis an isomorphism from BPto BQ.

4 The Impact of Pattern Morphism on Concept Lattices

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

C_G: tinf_DδX |X„Gu

forms a closure system inDand furthermoreCG:D|CGforms a complete lattice.

IfG pG,D,δqandH pH,E,εqeach is a pattern setup, then a pairpf,ϕqforms a pattern morphismfromG toH 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.

Theorem 3. Letpf,ϕqbe a pattern morphism from a pattern structureG pG,D,δq to a pattern structureH pH,E,εq.

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,„qand 2^H: p2^H,„qvia

α: 2^GÑ2^H,XÞÑf X and

α : 2^HÑ2^G,Y ÞÑf¹Y.

Further letϕ denote 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|d„δgu

: 2^HÑE,ZÞÑinf_EεZ

:EÑ2^H,eÞÑ thPH|e„εhu

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. In particular,pf XqϕpXqholds for all X„G.

We receive the following diagram of adjunctions:

2

2

E

D

α

α

ϕ ϕ

l

For the following we recall that the concept lattice ofG is given byBG:BP and the concept lattice ofH isBH :BQ.

Then Theorem 1 yields that the quadruplepBG,BH,Φ,Φ qis an adjunction for Φ: BGÑBH,pX,dq ÞÑ ppϕdq,ϕdq

and

Φ : BH ÑBG,pZ,eq ÞÑ pf¹Z,pf¹Zqq.

2^G 2^H

E^op D^op

α

ϕ

BG Φ BH

By Theorem 2 the following hold:

(1) If f is surjective thenΦis surjective too.

(2) Ifϕis injective thenΦis injective too.

(3) If f is surjective andϕis injective thenΦis an isomorphism fromBGtoBH. Theorem 4. LetG pG,D,δqandH pH,E,εqbe pattern structure. And letG pG,CG,δqbe the pattern structure induced byG viaδ:GÑC_G, gÞÑδg. It follows BGBG. Further let pf,ϕq be a pattern morphism fromG to H. Then with the notation introduced in the previous theorem, the mapΦ fromBG toBH is residuated.

If f is surjective then so isΦ, ifϕ is injective then so isΦ. If f is surjective andϕ is injective thenΦis an isomorphism fromBGtoBH.

Definition 6. Therepresentation contextof a pattern structureG pG,D,δqw.r.t. a subset M of D is given byKpG,Mq: pG,M,Iqwith I: tpg,mq PGM|m„δgu.

Theorem 5. LetG pG,D,δqbe a pattern structure and let M be a subset of D.

The pattern structure associated with the representation contextKpG,Mqis given by H : pG,2^M,εqwithε:GÑ2^M,gÞÑM^Óδg where_M^Ód: tmPM|m„dufor all dPD.

In particular, the concept lattice ofKpG,Mqis given byBKpG,Mq BH.

Using the notation from the previous theorem,pid_G,ϕqis a pattern morphism fromG toH forϕ:C_GÑ2^M,xÞÑM^Óx. Furthermore, the mapΦ fromBG toBH BKpG,Mq is a residuated surjection. In case M is join-dense w.r.t.CG(that is,ϕis injective),Φis an isomorphism fromBGtoBKpG,Mq.

2

2

2

C

id

ϕ X

X

X X

X X

d _M^Ód

2^G 2^G

2^M CG

id

ϕ

BG Φ BKpG,Mq

Remark: Based on the paradigm of concept morphisms, the previous theorem extends and sheds new light on theorem 1 of [3]. We generalize the definition of representation context introduced in [3] by allowing an abitrary subsetMof patterns of the underlying pattern structureG as attribute set of the representation contextKpG,Mq. It then turns out thatKpG,Mqhas a formal concept lattice which is induced by a morphism onG. More explicitly, there is a morphism onG to the pattern structure ofKpG,Mqwhich induces a residuated surjection from the concept lattice ofG to the concept lattice of KpG,Mq. In caseM is join-dense w.r.t.G , the morphism between the concept lattices is an isomorphism (see also Theorem 1 of [3]). Our extension of the concept of repre- sentation context gives rise to various constructions of o-projections (as introduced in [2]) onG.

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.

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.

9. L. Lumpe, S. E. Schmidt (2015), Pattern Structures and Their Morphisms. CLA 2015, pp 171-179.