graphical representation of mv-algebra...

Graphical Representation of MV-algebra Pastings

Ferdinand Chovanec

Department of Natural SciencesArmed Forces Academy

SK-03101 Liptovský MikulášSlovak Republic

E-mail: [email protected]

Ferdinand Chovanec Graphical Representation of MV-algebra Pastings

Hasse diagrams

A very useful tool to a graphical representation of finite partiallyordered sets (posets) are Hasse diagrams.

A Hasse diagram of a finite poset S is an orientated graph(digraph) where objects, called vertices, are elements of S andedges correspond to the covering relation.

Vertices are usually drawn as points or small black circles andedges as lines going from lower to higher elements, i. e. edges areupwardly directed.

Hasse diagrams

Hasse diagram of orthostructures

xOrthoalgebras

xOrthomodular Posets

xDifference Posets (Effect Algebras)

xOrthomodular Lattices

xDifference Lattices

xBoolean Algebras

xMV-Algebras

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xOrthoalgebras

xBoolean Algebras

xMV-Algebras

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xOrthoalgebras

xBoolean Algebras

xMV-Algebras

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Greechie logics

A method of a construction of quantum logics (orthomodularposets and orthomodular lattices) making use of the pasting ofBoolean algebras was originally suggested by Greechie in 1971.

GREECHIE, R.J.: Orthomodular lattices admitting no states, J.Combinat. Theory 10 (1971), 119 – 132.

These quantum logics are called Greechie logics.In Greechie logics, Boolean algebras generate blocks with theintersection of each pair of blocks containing at most one atom.

A block is a maximal subset of mutually compatible elements(a Boolean subalgebra) of a quantum logics.

Greechie logics

Greechie diagrams

A Greechie diagram of a Greechie logic L is a hypergraph wherevertices are atoms of L and edges correspond to blocks (maximalBoolean sub-algebras) in L .

Vertices are drawn as points or small black circles and edges assmooth lines connecting atoms belonging to a block.

Greechie diagrams are basically condensed Hasse diagrams.

A Greechie diagram of a finite Boolean algebra enables toreconstruct this algebra. If a Boolean algebra A contains n atoms(the Greechie diagram of A consists of n vertices lying on oneline), then A is isomorfic to the power set of a set with nelements, thus the cardinality of A is 2n.

Greechie diagrams

A Greechie diagram of a Greechie logic L is a hypergraph wherevertices are atoms of L and edges correspond to blocks (maximalBoolean sub-algebras) in L .

Vertices are drawn as points or small black circles and edges assmooth lines connecting atoms belonging to a block.

Greechie diagrams are basically condensed Hasse diagrams.

A Greechie diagram of a finite Boolean algebra enables toreconstruct this algebra. If a Boolean algebra A contains n atoms(the Greechie diagram of A consists of n vertices lying on oneline), then A is isomorfic to the power set of a set with nelements, thus the cardinality of A is 2n.

Loop Lemma

One of the useful tools in order to construct interestingorthomodular lattices and orthomodular posets is Greechie’s LoopLemma. Loop Lemma gives the necessary and sufficient conditionsfor Greechie logics to be an orthomodular lattice.

Loop Lemma (Greechie)A Greechie logic L is an orthomodular lattice if and only if Ldoes not contain neither a loop of order 3 nor a loop of order 4.

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Loop Lemma

One of the useful tools in order to construct interestingorthomodular lattices and orthomodular posets is Greechie’s LoopLemma. Loop Lemma gives the necessary and sufficient conditionsfor Greechie logics to be an orthomodular lattice.

Loop Lemma (Greechie)A Greechie logic L is an orthomodular lattice if and only if Ldoes not contain neither a loop of order 3 nor a loop of order 4.

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Stateless orthomodular lattice

Stateless orthomodular lattices

Later the method of the pasting of Boolean algebras wasgeneralized by many authors, mainly by

Dichtl, M.: Astroids and pastings, Algebra Universalis 18 (1984),380 – 385.

Navara, M., Rogalewicz, V.: The pasting constructions fororthomodular posets. Math. Nachrichten, 154(1991), 157-168.

Navara, M.: State spaces of Orthomodular Structures, Rend. Istit.Mat. Univ. Trieste, 31(2000), Suppl. 1, 143-201.

Navara, M.: Constructions of quantum structures, In: Handbook ofQuantum Logic and Quantum Structures: Quantum Structures.(Eds. - K. Engesser, D. M. Gabbay, D. Lehmanm) Elsevier 2007,335-366.

Short time after D-posets (effect algebras ) were discovered(1994), the attempts have been arose to generalize the method ofthe pasting in order to construct miscellaneous examples ofdifference posets and in order to study difference lattices admittingno states (probability measures).

These efforts were successful only after Riečanová proved thatevery lattice-ordered effect algebra (D-lattice) is a set-theoreticalunion of maximal sub-D-lattices of pairwise compatible elements,i. e. maximal sub-MV-algebras.

A method of a construction of difference lattices by means of anMV-algebra pasting was originally suggested inCHOVANEC, F.—JUREČKOVÁ, M.: MV-algebra Pastings, Inter.J. Theor. Phys. 42(2003), 1913-1926.

Construction of an MV-algebra pasting

What do we mean by an MV-algebra pasting?

Let S = {At : t ∈ T ,T is an index set } be a system of atomicσ -complete MV-algebras. By an MV-algebra pasting of thesystem S we understand a construction of a difference poset Pthat the following conditions are held.

(P1) There is a system S ∗ = {A ∗t : t ∈ T} of

sub-MV-algebras of P.

(P2) There is a bijection ψ from S onto S ∗ such thatthe MV-algebras At and A ∗

t = ψ(At) are isomorphicfor every t ∈ T .

(P3) P =⋃t∈T A ∗

What do we mean by an MV-algebra pasting?

Let S = {At : t ∈ T ,T is an index set } be a system of atomicσ -complete MV-algebras. By an MV-algebra pasting of thesystem S we understand a construction of a difference poset Pthat the following conditions are held.

(P1) There is a system S ∗ = {A ∗t : t ∈ T} of

sub-MV-algebras of P.

(P2) There is a bijection ψ from S onto S ∗ such thatthe MV-algebras At and A ∗

t = ψ(At) are isomorphicfor every t ∈ T .

(P3) P =⋃t∈T A ∗

By the expression at(A ) we denote the set of all atoms of anMV-algebra A and by |A| the cardinality of a set A.

Let A and B be atomic σ -complete MV-algebras. Let A and B befinite sets of atoms such that A⊆ at(A ), B ⊆ at(B) and|A|= |B|. We say that the sets A and B are isotropicallyequivalent, and write A∼τ B, if there is a bijection ϕ : A→ Bsuch that τ(a) = τ (ϕ(a)) for every a ∈ A.

By the expression at(A ) we denote the set of all atoms of anMV-algebra A and by |A| the cardinality of a set A.

Let A and B be atomic σ -complete MV-algebras. Let A and B befinite sets of atoms such that A⊆ at(A ), B ⊆ at(B) and|A|= |B|. We say that the sets A and B are isotropicallyequivalent, and write A∼τ B, if there is a bijection ϕ : A→ Bsuch that τ(a) = τ (ϕ(a)) for every a ∈ A.

Admissible system of MV-algebras

Let S = {At : t ∈ T ,T is an index set } be a system of atomicσ -complete MV-algebras At . We say that S is an admissiblesystem of MV-algebras (for a pasting), if the following conditionshold for arbitrary three MV-algebras A ,B,C ∈S .

(AS1) If A⊆ at(A ) and B ⊆ at(B) such that A∼τ B, thenat(A )rA 6= /0 and at(B)rB 6= /0. Moreover, ifat(A )rA= {a} or at(B)rB = {b}, respectively,then τ(a)> 1 and τ(b)> 1.

Let S = {At : t ∈ T ,T is an index set } be a system of atomicσ -complete MV-algebras At . We say that S is an admissiblesystem of MV-algebras (for a pasting), if the following conditionshold for arbitrary three MV-algebras A ,B,C ∈S .

(AS1) If A⊆ at(A ) and B ⊆ at(B) such that A∼τ B, thenat(A )rA 6= /0 and at(B)rB 6= /0. Moreover, ifat(A )rA= {a} or at(B)rB = {b}, respectively,then τ(a)> 1 and τ(b)> 1.

(AS2) If A⊆ at(A ), B ⊆ at(B), C ⊆ at(C ) such thatA∼τ B and at(A )rA∼τ C , then there is anMV-algebra D ∈S and a subset D ⊂ at(D) suchthat at(B)rB ∼τ D and at(C )rC ∼τ at(D)rD.

Let S = {At : t ∈ T} be an admissible system of MV-algebras.We choose two sets A and B from every pair of MV-algebrasAt ,As ∈S such that A⊂ at(At), B ⊂ at(As) and A∼τ B. Thechoice of isotropically equivalent pairs subsets of atoms allows todefine the relation ∼ on

⋃t∈T At in the following way.

(1) 0At ∼ 0As and 1At ∼ 1As whenever A= /0 and B = /0.

(2) If x ,y ∈At then x ∼ y if and only if x = y .

(3) If x ∈At and y ∈As and A= {a1,a2, ...,an},B = {b1,b2, ...,bn}, then x ∼ y wheneverx =

⊕ni=1 piai and y =

⊕ni=1 pibi , where bi = ϕ(ai ),

pi ∈ {0,1,2, ...,τ(ai )} for i = 1,2, ...,n.

(4) If x ∼ y then x⊥ ∼ y⊥.

[x ] = {y ∈⋃t∈T At : y ∼ x} (equivalence class determined by x)

P = {[x ] : x ∈⋃t∈T At} (quotient set)

A ∗t = {[x ] : x ∈At}

P =⋃t∈T A ∗

[x ]≤ [y ] if and only if there is an MV-algebra Ar ∈S andelements u,v ∈Ar such that u ∈ [x ], v ∈ [y ] and u ≤r v .

If [x ]≤ [y ] then we define

[y ] [x ] = [v r u].

0P = [0At ], 1P = [1At ],

(P,≤,1P ,0P ,) is a D-poset (an MV-algebra pasting in thesense of previous definition)

A ∗t = {[x ] : x ∈At}

P =⋃t∈T A ∗

[y ] [x ] = [v r u].

0P = [0At ], 1P = [1At ],

A ∗t = {[x ] : x ∈At}

P =⋃t∈T A ∗

[y ] [x ] = [v r u].

0P = [0At ], 1P = [1At ],

A ∗t = {[x ] : x ∈At}

P =⋃t∈T A ∗

[y ] [x ] = [v r u].

0P = [0At ], 1P = [1At ],

A ∗t = {[x ] : x ∈At}

P =⋃t∈T A ∗

[y ] [x ] = [v r u].

0P = [0At ], 1P = [1At ],

For a graphical representing of MV-algebra pastings we can usealso Greechie diagrams. In this case we denote a vertex of theGreechie diagram in the form a(τ(a)), where a is an atom and τ(a)is its isotropic index. Then a finite MV-algebra is uniquelydetermined by its Greechie diagram.

Let A = {(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1),(0,1,1), (1,1,1)} and B = {(0,0),(1,0),(0,1),(1,1),(2,0),(2,1)}.If an ordering and a difference of comparable elements are definedcoordinately, then A becomes a Boolean algebra and B becomesan MV-algebra.Denote 0A = (0,0,0), a= (1,0,0), b = (0,1,0), c = (0,0,1),a⊥ = (0,1,1), b⊥ = (1,0,1), c⊥ = (1,1,0), 1A = (1,1,1).0B = (0,0), e = (1,0), d = (0,1), e⊥ = (1,1), 2e = (2,0),1B = (2,1).

For a graphical representing of MV-algebra pastings we can usealso Greechie diagrams. In this case we denote a vertex of theGreechie diagram in the form a(τ(a)), where a is an atom and τ(a)is its isotropic index. Then a finite MV-algebra is uniquelydetermined by its Greechie diagram.

Let A = {(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1),(0,1,1), (1,1,1)} and B = {(0,0),(1,0),(0,1),(1,1),(2,0),(2,1)}.If an ordering and a difference of comparable elements are definedcoordinately, then A becomes a Boolean algebra and B becomesan MV-algebra.Denote 0A = (0,0,0), a= (1,0,0), b = (0,1,0), c = (0,0,1),a⊥ = (0,1,1), b⊥ = (1,0,1), c⊥ = (1,1,0), 1A = (1,1,1).0B = (0,0), e = (1,0), d = (0,1), e⊥ = (1,1), 2e = (2,0),1B = (2,1).

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u[c](1)=[d](1)

u[e](2)

u[0A ]=[0B]

u[c]=[d]

u[2e]=[c⊥] u[b⊥] u[a⊥]

u[1A ]=[1B]

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Generalized Greechie diagrams

Greechie diagrams are useful only in the case if the intersection ofblocks contains a small number of atoms. If it be to the contrary,we suggest to use a generalized Greechie diagram.

Let P be an MV-algebra pasting. A generalized Greechie diagram(a GG-diagram for short) is a hypergraph (V ,E ), where V (a setof vertices) is a system of finite and pairwise disjoint subsets ofat(P) and E is a system of nonempty subsets of V (edges) suchthat

⋃E = V .

Vertices of a GG-diagram we draw as small circles and edges assmooth lines connecting sets of atoms belonging to a block.

⋃E = V .

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Loops in MV-algebra pastings

Let P =⋃t∈T A ∗

t be a pasting of an admissible system ofMV-algebras S ={At : t ∈ T}. Let A ∗

0 ,A∗1 , . . . ,A

∗n−1 be a finite

sequence of blocks in P.Denote

Vi+1= at(A ∗i )∩at(A ∗

i+1)rn−1⋃

k=0;k 6=i ,i+1at(A ∗

k ), i = 0,1, . . . ,n−1 (mod n)

W =n−1⋂k=0

at(A ∗k ).

(1) A finite sequence A ∗0 ,A

∗1 , . . . ,A

∗n−1 is said to be a loop of

order n, n ≥ 3, if the following conditions are fulfilled.(L1) Vi 6= /0 for every i = 0,1, . . . ,n−1.

(L2)(at(A ∗

i+1)∩at(A ∗j )r

⋂n−1k=0;k 6=i+1,j at(A

∗k ))= /0,

for every i = 0,1, . . . ,n−1 andj /∈ {i , i +1, i +2} (mod n).

(2) Atoms belonging to the sets Vi , i = 0,1, . . . ,n−1, are callednodal vertices, and atoms belonging to the set W are calledcentral nodal vertices of the loop of order n.

(3) A loop of order 4 is called an astroid ifat(A ∗

i ) = Vi ∪Vi+1∪W for every i = 0,1,2,3 (mod 4).

Comments:(a) It is easy to verify that Vi ∩Vj = /0 for every i 6= j .(b) All atoms of an astroid are the only nodal vertices.

Astroids

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Loop Lemma

(1) We say that a loop of order 3 is unbound in an MV-algebrapasting P if there is no block in P containing all its nodalvertices.

(2) We say that a loop of order 4 is unbound in an MV-algebrapasting P if there is no block and no astroid in P containing allits nodal vertices.

TheoremAn MV-algebra pasting P is lattice ordered if and only if Pcontains no unbound loops of order 3 and 4.

Loop Lemma

Examples

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