On
the relation of Fuzzy subsets, Postean and Boolean lattices .
The ë-rainbow
lattices. Transfinite Fuzzy subsets.
By Dr Constantine E. Kyritsis
Department of Mathematics and Computer Science
§ Ï. Introduction. In this paper it is attempted to relate
the definitions of the fuzzy subset lattices with the other classical
definitions of Postean and Boolean lattices in the context of Universal
algebra. In particular it is obtained a kind of equivalence (mutual isomorphic representations)
of the fuzzy subsets, Postean and Boolean lattices. (corollaries
28,29,30.).This unification is worked out with the axiomatic introduction of
the ë-rainbow lattices (§1). In order to obtain
these results, it is used the extended Stone's representation theorem (theorem
1). The way that this theorem is formulated makes use of the extended
Birkhoff-Stone theorem (corollary 21).
It is given new and more enlightening and simpler proofs for both theorems.
They are discussed partial abelian semigroups, abelian groups, rings and
modules that are also lattices. It is Introduced the concept of compatibility of the order relation and the
algebraic structure. The previous equivalence theorems have important
consequences in a kind of equivalence, up-to-lattices of the Fuzzy, 3-valued
and 2-valued Logic. It is possible obtain the fuzzy logic as a general
Logic (according to [J. Mesenquer 1987]). By introducing an
ordering it is possible to investigate
the relation of a general Logic, the Fuzzy, the 3-valued and the 2-valued
Logic.
In the paragraph §2 is introduced the
transfinite real, complex, quaternion Fuzzy subset lattices. They can be
considered as a kind of Arithmetisation of the abstract ë-Fuzzy
lattices. They have special good properties that can be understood only by a
familiarity with the transfinite real numbers.
§1 The ë-rainbow (or fibber) lattices and the ë-Fuzzy subsets
lattices.
As it is known the category of Boolean
lattices has been extensively studied, mainly
because of its applications in Logic.
The attempts to give alternative formulations of Logic led to other categories
of lattices, like the Postean lattices (see e.g.[Carvallo
M.1968],[Lukasiewicz,J1920],[Post E.L. 1921]), or the
lattices of Fuzzy
subsets (see e.g.,[Zadeth,L.A 1965]). A closer study
of the Boolean, Postean Fuzzy -subsets, modular-
complemented lattices leads naturally to a broad category of lattices
which we define
axiomatically in this paragraph and
we call rainbow
or fibber lattices. In
conceiving the idea of rainbow
lattices I was partly influenced at least by the terminology, of the Kantian
ideas of transcendental analytic and synthetic logic and of Aristotle’s term of
“colors of the word”. As we shall see transfinite
numbers appear also.
Remark: If a lattice has minimum and maximum elements we denote them
with 0,1 respectively.
We are now in a position to define the
rainbow (or fibber) lattices: Let an non-empty order type which we denote by ë. Let
a lattice, which we denote by L such that:
The ë-rainbow
(or fibber) axioms:
A_{0}: The order type ë is
a sublattice of L and the L has 0 and
1.
A_{1}: If áÎë á¹0
and for a xÎL it holds xÙa=0 then x=0.
A_{2}: If á Îë, á¹1
and for a xÎL holds xÚa=1 then x=1.
The spectral axioms:
For every xÎL there is a ë-family of elements of L {x_{á}|
áÎë}
which, is called the ë-spectrum
of the x, such that:
B_{0}: _{} The previous representation is called the ë-spectral
expansion of x.
B_{1}: If á_{1}¹á_{2 }for
two ë-indices of the spectrum of x, then x_{á}_{1}Ù x_{á}_{2 }=0
B_{2}: For the ë-spectrum of x holds _{}.
Definition 1. A
Lattice that satisfies the axioms A_{0}, A_{1}, A_{2}, B_{0}, B_{1}, B_{2 }as above, is
called a ë-rainbow (or ë-fibber)
llattice.
Remark 2: If the ë is the order type of an
ordinal number á, we write simply a-rainbow
lattice. If ë=3={-1<0<+1}
and the L is furthermore distributive we
get the definition of a
Postean lattice (see e.g.[Carvallo ,M.1968],[Lukasiewicz,J.1920],[Post, E.L.
1921]). In other words a Postean lattice is defined as a distributive 3-rainbow lattice. We
shall prove that the Fuzzy-subsets lattices are ë-rainbow lattices .We shall describe in the first a
way to construct ë-rainbow lattices from ë-order
types.
Remark. Let us suppose that the order type
is at the same time a commutative
partial semigroup such that the
order and the operations are
compatible: In other words if
we denote by i the identity mapping and for two elements a, b the sums i
(a)+i(b), i(aÚb)+i(aÙb) exist then it holds that i(aÚb)+i(aÙb)=i(a)+i(b).
In other word, again the identity mapping is a (partial semigroup valued)
lattice valuation .Then a ë-rainbow lattice on such a ë is
called rainbow lattice with algebraic fibber. If the
partial semigroup ë, is a category, commutative group, ring, field, module, or
vector space, then we put this
term before the term algebraic.
Lemma 3. The
(direct) product _{} in the category of lattices of the order type ë
with extremes 0,1, X-times where X is
an arbitrary set, is a distributive ë-rainbow
lattice.
Proof: The
order type ë is contained as a sublattice in the _{}, as the element with all the coordinates
equal.
Thus the axioms A_{0}, A_{1}, A_{2} are satisfied .We define as the ë-spectrum
of an element x of _{}the elements {x_{á}|
áÎë} such that the x_{á} is the element with all the coordinates equal to 0
except at the places where the coordinates of x are
equal to a where it is equal, to 1. It
is direct that the axioms B_{0}, B_{1}, B_{2} are satisfied. The order type ë is
a distributive lattice. Obviously
the same holds for the (direct) product _{} Q.E.D.
Remark 4. The ë-rainbow lattices _{} with extremes 0,1,
(in
other words as in lemma 3 ) are the ë-Fuzzy
subsets lattices
Corollary 5. The ë-Fuzzy-subsets lattices are distributive, ë-rainbow
lattices.
Proof: A ë-Fuzzy-subset
lattice on a set X is exactly the (direct), product _{} in the category of lattices ;Where the ë
is, an order type .Thus we apply the previous Lemma Q.E.D.
Proof :1Direct from the definitions and that ë_{1 }£_{ }ë_{2}. Q.E.D.
Lemma 7. A lattice is complemented
if and only if it is a 2-rainbow
lattice.
Remark 8. A lattice is called complemented if it has extremes, 0,1 and for
any element x there is an element x' such that, xvx'=1 and x^x'=0 .The x' is
called the complement of x .(See also [v Neumann J.1960] part 1 ch 1 axiom v pp
2 or the paper with the same title in his collected works ) .It is not
postulated uniqueness of the complement of an element .
Proof
: The order type ë is the two element set
{0,1} thus the, order type of 2. Thus the axioms A_{0}, A_{1}, A_{2}
are satisfied .As the
{0,1}spectrum of any element x we define the two element set {x_{0}, x_{1} } with x_{0}=x' and x_{1}=x
.By x' we denote the complement of x . It holds by the definition of the
complement that (0^x_{0})
v(1^ x_{1} )=(0^x')v(1^
x)=0vx=x ;and x_{0}^x_{1} =x'^x=0, x_{0} vx_{1}
=x' vx=1.
Thus the axioms B_{0}, B_{1}, B_{2} are satisfied. Conversely
in every 2-rainbow lattice
the x_{0} can be taken as a complement of x=x_{1}. Q.E.D.
Corollary 9. A lattice is a Boolean lattice if and only if it is, a
distributive 2-rainbow lattice.
Proof: Direct from the previous lemma and that every Boolean, lattice is defined as a complemented (distributive) lattice. Q.E.D.
§2 .The transfinite real, complex,
quaternion Fuzzy subsets
and lattices.
Remark 10.
In the next we introduce the transfinite real, complex,, quaternion Fuzzy
subsets .We shall examine at first the elementary cases of the real, complex
and quaternion fuzzy subsets. In the definition of the ordinary [0,1]-fuzzy
subsets (see e.g.[Zadeth,,L.A.1965]) what matters is the interval [0,1] as an
order type and not what numbers are the extremes of it. Let us consider the
2-point compactification of the linearly ordered field of the real numbers _{}, where the -¥, +¥ are the minimum and maximum of _{}. As it is known the _{}is order isomorphic order
type to the order type [0,1]. Therefore the [0,1]-Fuzzy subsets can be considered
as _{}-Fuzzy subsets. The latter
form is some how more convenient as we can make use of the algebraic structure
of the field _{}. Furthermore negative values of the index or characteristic
function of a fuzzy subset can appear.
It is very natural to substitute the linearly ordered
(commutative) field _{}with the field of the
complex numbers C or the (non-commutative) field of the quaternion numbers H.
(See also [v.Weizacker C.F.-Sheibe,E. Sussmann G. 1958]. Actually we have to use the compactifications of them _{}. Thus the elements of the products in the category of sets _{}, where the x is an arbitrary set, are the real ,complex and
quaternion Fuzzy subsets of x .The norm of the characteristic functions of them
give real Fuzzy subsets. But the advantage of the complex and quaternion Fuzzy
subsets is that besides their real norm they have respectively a phase e^{i}^{¶}, e^{q}^{¶} (where i is the
imaginary unit and the q is a unit vector of the Euclidean space R^{3}).
The way to define the transfinite real, complex and quaternion Fuzzy subsets is
plausible and formalises a non-Archimedean phased Fuzziness.It is to be
understood that the next arguments do not have to depend on the [Conway
J.H.1976],[Robinson A. 1966].It can be proved that all the three different
techniques of the transfinite real numbers,
the surreal numbers and the ordinal real numbers give by inductive limit
or union the same class of numbers known already as the class No. This class
can be called the totally ordered, finitary Newton-Leibniz realm of numbers.
Till the moment that this shall be understood we can be content
with the transfinite real numbers
in [Glayzal A.1937].Although logically not necessary, any additional references
to the ordinal real numbers shall be in parenthesis and shall not affect the
arguments. The author believes that the Ordinal real numbers shall be
indispensable, in the future, for numerical and quantitative applications of
mathematics to other sciences.
The transfinite real numbers were
defined at first by A.Glayzal in 1937(!) (see [Glayzal A.1937] ). His technique
was what it is now known as formal power series fields. His main idea was to
define by formal power series fields,
Archimedean complete (although non-Archimedean) linearly ordered fields. The
concept of non-Archimedean order is known in the process management of
operating systems in computer science and in operation research (Goal
programming) as pre-emptive prioritisation.
(It it can proved that they are fundamental (Cauchy)
complete (in the order topology ) and Dedekind complete. I repeat that the
reader can be content with their definition in [Glayzal A.1937] which we repeat
concisely .Let any order type which we denote by ë. Let
the Lexicographic product of ë-copies of the field of the real numbers R, which we denote by _{}. Its elements are the elements of the Group-product _{}which have zero all the components except on a well-ordered
subset of ë. This subset is called as usually the support of the element.
The group _{}is a linearly ordered group in the lexicographic ordering .The formal power series field _{} is defined as the transfinite
real numbers of Archimedean base ë. Let us denote by R (R (ë) ) the real closure of
the transfinite real numbers ; it can
be obtained by adjunction of the square roots of its
positive elements But the R (ë) is a formal
power series field, thus it contains
the n-roots of its positive elements (see [v.Neumann] B.H. 1949] 4.91 Corollary
pp 211). In other words it is Pythagorean complete. Thus R (R (ë) ) = R (ë).
Consequently its extensions of degree 2 with the imaginary
unit C (ë) = R (ë) (i) is also algebraically closed .The algebraically closed and
formal power series field C (ë) we call transfinite complex numbers of Archimedean
base ë . It holds obviously that _{}. Similarly the field that is obtained by
the quaternion extension of R (ë) ,which is the field _{} is
called the transfinite quaternion numbers of Archimedean base ë. Let
us consider the extensions of them _{}
where the -¥, +¥ are the minimum and maximum of the order type _{}.
Definition 11. The products _{} in the category of sets,
where the X is an arbitrary set ,are called the transfinite, real ,complex, quaternion Fuzzy subsets of X.
Remark 12. The
norm of the transfinite real ,complex ,quaternion
numbers of some Archimedean base ë , is defined as in the
minimal case of the real,
complex ,and quaternion numbers. Only
that the norm is not a positive real number but a transfinite positive real
number. By taking the norm , the transfinite real, complex, quaternion Fuzzy subsets
define _{}-fuzzy subsets.
Such (real) Fuzzy subsets we call transfinite Fuzzy subsets. They are special
Fuzzy subsets, since the order types _{}have special good
properties; (fundamental completeness in the order topology, Dedekind completeness , they contain initial segments of the
ordinal numbers e.t.c.).
The applications of the (real) transfinite Fuzzy subsets are
as the applications of the general Fuzzy subsets and even better.
Remark.
By the way that the transfinite real numbers are defined (over an arbitrary order type which is its Archimedean base)
it is clear that any order type ë is order embeddable
in some field of transfinite
real numbers .Therefore after lemma 6 every ë-Fuzzy subsets lattice is isomorphic with the a sublattice of some
transfinite real Fuzzy subsets
lattice.
Corollary (The transfinite Fuzzy subsets lattice reduction )
Any ë-Fuzzy subsets lattice is isomorphic with a sublattice of some transfinite (real) Fuzzy subsets lattice.
Remark. It can be proved that any
order type can be constructed by the ordinal numbers. This constitutes a
genuine and better Arithmetisation of the ë-Fuzzy subsets).
§3 Rings that are also lattices. Auxiliary rings.
As it is known J.v.Neumann has studied the orthomodular 2-rainbow
lattices that appear as lattices
of projections in Hilbert spaces. He has also introduced and studied a
special category of modular 2-rainbow lattices that he called continuous
geometry (see [v.Neumann,J. 1960]). He proved that such lattices admit
valuations in the real numbers .He proved that the modular 2-rainbow lattices
have an up-to-isomorphism representation as lattices of ideals of rings. The
ring that corresponds to every such lattice is called auxiliary ring (see
[v.Neumann J.1960] part ii). This technique is not little laborious.
In the case of the Boolean lattices the auxiliary ring is
again the Boolean algebra and the ring operations are x+y=(x^y')v(x'^y) and
x.y=x^y .Any element of the Boolean lattice corresponds to the principal ideal
that it defines . The principal lattice-ideals and the principal ring-ideals
coincide. The correspondence is a
lattice isomorphism. The present representation theorem gives also a
result of the same type. In other words an auxiliary ring appears also. Thus it
can be also considered as a J.v.Neumann- type representation.
Definition 13.
Let us denote a lattice by L and a ring by R. The ring R is called auxiliary to
the lattice L if there is an isomorphism of L with a lattice of ideals of R.
Such an isomorphism is called a J.v.Neumann-type
representation.
Remark 14: By
the Stone's representation theorem we deduce that every Boolean lattice is
isomorphic with a sublattice of the Power- set P(X) which is also a lattice of a set X .Considering the
characteristic functions of the subsets of X we deduce that every Boolean
lattice is isomorphic with a sublattice of the abelian Æ_{2}-linear algebra _{}. The product is considered in the category of rings and
simultaneously in the category of lattices .The Galois field Æ_{2} is considered as a lattice with the ordering 0<1.
It might be interesting to have a closer look to modules and
rings that are also lattices (see also remark after remark 2).
Let a ring R and an lattice ordering on it .We do not suppose
any compatibility of the ordering and the ring operations.
E.g. we may consider the rings Z_{n} with the linear ordering
0<1/_{ð}_{-1}<1/_{ð}_{-2}<…<1/_{2}<1.
Or we may consider any linearly ordered ring e.g. the field
real numbers.
The (direct) product _{} over a set of indices
X simultaneously
in the category of rings , and of lattices is a ring which
is also a lattice .In the previous examples the _{}are Z_{n}-modules and rings which are also distributive n-rainbow lattices.
If n=p where p is a prime natural number then the
distributive p-rainbow lattice _{}is also a Z_{p}-linear algebra.
In these commutative Z_{p}-linear algebras holds that x^{P}=x and px=0 (it is of, characteristic n) ;which is a
consequence of the Fermat theorem for the Galois fields Z_{p}.
The identity function has the property that xvy + xÙy = x+y.
Thus it is a Lattice valuation with values in the
commutative semigroup
_{}. For n=2 or 3 the relation of the lattice-operations and the
ring operations are:
éa.b if a≠b éa+b
-a.b if a≠b
a^b = ëa if a=b
avb= ëa if a=b .
The lattices _{}p=2,3 have as auxiliary rings again the _{}.
To every element corresponds the principal ring-ideal that
it defines. The principal lattice-ideals and the principal ring-ideals
,coincide.
§4 The inverse of the Birkhoff-Stone theorem in lattices
Definition 15. A
set , that we denote by È , of
congruencies in a lattice,
that we denote by L, is said that it is separating iff for any two elements x,yÎL with x ≠y there is a congruence ¶ÎÈ such
that x≠y mod¶.
Remark 16: If
the lattice L has minimum, that
we denote by 0, then as it is known to every congruence ¶ corresponds an ideal that we denote by É_{¶}, which is the equivalence class of 0. Thus we have also the
next definition :
Definition 17
Let a lattice, be denoted by L, and a set of ideals of it, be denoted by I. It is said that the I is a separating set of ideals iff
for any two x, yÎL with x≠y there is an ideal iÎI
such that xÎi and yÏi.
Definition 18.Let
a lattice, be denoted by L. The set of its prime ideals we denote by SpecL and
we call the spectrum of the lattice L.
Definition 19. A
lattice, which is denoted by L, is said that it is with separating spectrum iff
the set of ideals SpecL is separating in the L.
Theorem 20.
(The extended Birkhoff-Stone theorem)
A lattice is with
separating spectrum if and
only if it is
distributive .
Proof: see [Sirkoski,R 1969] Appendix 6 §13 first page of the paragraph
Q.E.D.
The Stone's representation theorem makes use of the fact
that the spectrum of a Boolean
lattice coincides with
the set of its
maximal ideals and that and the Birkhoff-Stone theorem which
gives that any distributive lattice
is with separating
spectrum .
The Stone's representation theorem can be extended
to any
lattice with separating spectrum .
Theorem 21.
(The extended Stone's isomorphic representation theorem ).
Let us denote by L a lattice with separating spectrum.
There is an
isomorphism of L with a lattice of sets.
Proof: See [Birkhof,G 1967] ,or [Gratzer,G 1979]
,or[Sikorski,R 1969] or any
comprehensive book about
lattices. Q.E.D.
Corollary 22.
(J.v.Neumann-type isomorphic representation )
Every distributive lattice has an auxiliary (abelian) ring.
Proof:
The distributive 2-rainbow lattices _{}by the last remark of §3 have auxiliary ring the same set as
a product abelian ring.
Q.E.D.
The next corollaries are results about the relation of the
Boolean, Postean, and Fuzzy-subset lattices.
Corollary 23. (The Z_{2} reduction)
Every Postean and Fuzzy-subset lattice is isomorphic to a
lattice of sets and has an auxiliary ring.
Corollary 24. Every
distributive ë-rainbow lattice is isomorphic with a sublattice of a ë’-rainbow
lattice _{}, for every order type ë’ with
extremes, in other words with a sublattice of a ë’-Fuzzy subsets lattice.
Proof: We apply the theorem 20 and the lemma 6 .The Z_{2}
2-rainbow lattice is the minimal (or trivial ) rainbow lattice.
Corollary 25. (The ë-Fuzzy representation of distributive ë-rainbow lattices)
Every distributive ë-rainbow
lattice is isomorphic with a sublattice of a ë-Fuzzy lattice.
Corollary
Every distributive ë-rainbow lattice is isomorphic with a sublattice of a transfinite (real) Fuzzy subset lattice.
Proof: We
apply the transfinite Fuzzy reduction of the Fuzzy subsets lattices of §2 and
the previous lemma.
Corollary 26.
(The Z_{3} reduction). Every distributive lattice and in particular
every Fuzzy-subset lattice is isomorphic with a sublattice of a Postean lattice
of type _{}.
Corollary 27.
(The Fuzzy-reduction)
Every distributive lattice isomorphic with a sublattice of a Fuzzy-subset lattice
Corollary 28. (The
Z_{2}↔Fuzzy equivalence)
Every Boolean lattice is isomorphic with a sublattice of a
Fuzzy-subset lattice and every Fuzzy-subset lattice is isomorphic with a
sublattice of a Boolean lattice.
Corollary 29.
(The Z_{2}↔Z_{3 }equivalence)
Every Boolean lattice is isomorphic with a sublattice of a
Postean lattice and every Postean lattice is isomorphic with a sublattice of a
Boolean lattice.
Corollary 30.
(The Z_{3}↔Fuzzy equivalence)
Every Postean lattice is isomorphic with a sublattice of a
Fuzzy-subset lattice and every Fuzzy-subset lattice is isomorphic to a
sublattice of a Postean lattice.
Corollary 31. (The
equivalence up-to-lattices of the Fuzzy, 3-valued and 2-valued Logic.)
The Fuzzy, 3-valued, 2-valued Logics are up-to-lattices
equivalent in the sense that there are mutual isomorphic representations of the
logical (lattice) orders of them.
(See also [A.I.Appudia etc 1977],[J.Barwise,S Feferman
1985],[Carvallo ,M.1968],[Conway J.H.1976],[Mesenqeur J.1989])
Remark 32. We
remind to you that by the extended Stone's representation theorem for any
distributive lattice we get as a corollary that:
Every partially ordered set has a monomorphic image in a
lattice of sets (relative to inclusion ).
Every partially ordered set has a monomorphic image in the
full binary tree.
This is a very useful result that is not mentioned widely.
It can be also considered as an extension of the Stone's
representation theorem, in two directions: a) to every (partially)ordered set
b) the representation is not order-isomorphism but order-monomorphism
(isotonous mapping ).
About its proof: As it is known, it holds a
"linearisation" theorem for any (partially) ordered set :Every
(partially) ordered set has an order-monomorphic image in an order type .The
order types are distributive lattices. Thus by the extended Stone's
representation theorem every (partially) ordered set has an order-monomorphic
image in a lattice (relative to inclusion) of sets.
About the second assertion: The lattice of sets that was
mentioned in the previous assertion is a Boolean lattice of the type_{}. The lattice direct-product ordering of it has by the
identity mapping an order-monomorphic image in the set _{}={0,1}^{á} and in the lexicographic ordering; where the a is the cardinality of X, thus an initial ordinal
number.
The {0,1}^{á} is
a level of the full binary tree D_{á} of height a.
The ordering of the level can be taken to be the
lexicographical ordering Q.E.D.
Acknowledgments. I
would like to thank professor Kirby Baker (Mathematics Department of the UCLA)
for the interest that he showed for the ë-rainbow
lattices and his valuable remarks.
Also professor G.Moschovakis (Mathematics Department of the
UCLA) for giving to me the opportunity to visit the Mathematics Department of
the UCLA.
Bibliography
Abbot, J.C. 1969 |
Sets,
Lattices, and Boolean Algebras, Allyn and Bacon, 1969. |
A.I. Appudia, N.C.A., Da Costa R. Ghuaqui 1977 |
Non-classical
Logics, Model Theory and Computability North-Holland
1977. |
J. Barwise, S. Feferman 1985 |
Model Theoretic
Logics especially
chapter II Extended
Logics by H-D Ebbingaus Springer 1985. |
Birkhoff, G. 1967 |
Lattice
theory Amer. Math.
Soc. Coll. Publ., Vol., XXV,
Providence, 1967. |
Birkhoff, G. – MacLane, S. 1977 |
A survey of
Modern Algebra. Macmillan
Pub, C. 4-eddition 1977. |
Birkhoff G. – v. Neumann, J. 1936 |
The logic of
Quantum Mechanics. Annals of
Mathematics 37, pp 823-843 (1936) reprinted in J, v. Neumann
Collected Works vol. 4 pp 105-125. |
Carvallo, M.
1968 |
Logique à trios valeures
logique à seuil Collection de Mathematiques
Economiques Gauthier-Villars Paris 1968. |
Cohn P.M.
1965 |
Universal
Algebra Harper-Row-Weatherhill
P.C. 1965. |
Conway J.H.
1976 |
On numbers
and games. Academic
press 1976. |
H-D Ebbinghaus J. Flum W. Thomas 1984 |
Mathematic
Logic Springer
1984. |
L. Fuchs 1963 |
Partially
ordered Algebraic Systems Addison-Wesley Pergamon
press 1963. |
Glayzal A. 1937 |
Transfinite
real numbers Proc of the
Nat. Acad of scien. 23 (1937) pp 581-587). |
Grätzer, G. 1978 |
General
Lattice Theory, Academic
Press, 1978. |
Grätzer,
G. 1979 |
Universal Algebra Springer V. 1979. |
Lukasiewicz, J. 1920 |
O logice
trojwartisciowej (on three-valued logic) Ruch
Filozoficzny 5, (1920) pp 169-171 reprinted in J. Lukasiewicz, Selected
Works L. Borkowski North-Holland Pub.C. 1970 pp 87-88. |
MacLane S. 1971 |
Categories
for the Working Mathematician. Springer V. 1971. |
Neumann B.H. 1949 |
On ordered
division rings Transactions
of the A.M.S. 60 (1949) pp
202-252. |
von Neumann, J. 1960 |
Continuous
Geometry Princeton
University Press 1960. |
Piron
Constantin 1964 |
Axiomatique Quantique Helvetica Physica Acta 37 (1964) pp 439-468. |
Post, E.L. 1921 |
Introduction
to a general theory of elementary propositions Amer. J.
Math, vol. 43, (1921) pp 163-185. |
Putnam, H.
1957 |
Three-valued
logic Philosophical
studies 8. (1957) pp 73-80. |
Rasiowa Helena 1974 |
An Algebraic
Approach to non-classical Logics. North-Holland 1974. |
Rieger Ladislav 1967 |
Algebraic
methods of mathematic Logic. Academic 1967. |
Robinson A. 1966 |
Non-standard
Analysis North-Holland
1974 (1966). |
Rosser J.B.-Turkuette, A.R. 1952 |
Many-Valued
Logic North-Holland
Publ, C. 1952. |
Manin Y.I. 1977 |
A course in
Mathematical Logic. Springer
1977. |
J. Mesequer 1989 |
General
Logics In the
"Logic Colloquium 1987" Studies in Logic
v. 129 North Holland
1989. |
Sikorski, R. 1969 |
Boolean Algebras Springer Verlag, 1969. |
Springer Publications 1987 |
Ù-bibliography in Mathematic Logic
vol I: Classical Logics vol II: Non-classical Logics vol
III : Model Theory vol IV: Proof
Theory constructive mathematics. |
v. Weizsäcker C.F.-Sheibe, E. Sûssmann G. 1958 |
Koplementarität und Logik Zeitschrift für Naturforschung 13
a, pp 705-721 (1958). |
Zadeth, L.A. 1965 |
Fuzzy sets Information
and Control 8, 338-353
(1965). |
Key Words
Lattices,
general logic
Boolean lattices, logic
Postean lattices, logic
Fuzzy-subsets lattices, logic
transfinite real numbers
Note on the relation of the Fuzzy
subsets, Postean,
Boolean lattices. The ë-rainbow lattices.
The transfinite (real, complex,
quaternion) Fuzzy subsets.
By Dr Constantine E. Kyritsis
Abstract
In this short note the main results
are:
1) They are introduced axiomatically
the ë-rainbow lattices. It is proved that
the ë-Fuzzy subsets lattices are
distributive ë-rainbow lattices and other
relations of these two classes of lattices.
2) They are introduced the
transfinite real, complex, quaternion. Fuzzy subsets. They can be considered as
a kind of Arithmetisation of the abstract ë-Fuzzy subsets lattices. The transfinite real Fuzzy
subsets are special ë-Fuzzy subsets with good properties.
3) It is proved a mutual isomorphic
representation of the Boolean (2-valued logic) Postean (3-valued logic) and
Fuzzy-subsets (Fuzzy-logic) lattices.
extremes 0,1 of the lattice. We define the function: _{}by h(x)=a_{x} where a_{x} (t) = _{}.
We shall prove that the h is a lattice isomorphism of L and
its range _{}.
a) We shall prove that h (xÙy)=h(x)Ùh(y).
Let h(x)=a_{x} and h(y)=a_{y}.
Let us suppose that xÏt and yÏt
for some tÎ SpecL. Then a_{x}=1=a_{y}. If xÙyÎt and because the t isa prime ideal we deduce that either xÎt or yÎt,
contradiction. Thus xÙyÏt, hence a_{x}_{Ù}_{y}=1 and a_{x}_{Ù}_{y}=1=1Ù1=a_{x}Ùa_{y}. The other case is that either xÎt or yÎt.
Then either a_{x}=0 or a_{y}=0. In both cases, because the t is
an ideal xÙyÎt and thus a_{x}_{Ù}_{y}=0;
and a_{x}_{Ù}_{y}=0=0Ù(0
or 1) = a_{x}Ùa_{y}.
b) We shall prove that h(xvy)=h(x)vh(y). Let us suppose
that xÎt and yÎt
for some tÎSpecL. Then a_{x}(t)=1=a_{y}(t). If xvyÎ and because the t is an ideal we deduce that x, yÎt, which is a contradiction.
Thus xvyÎt
and a_{x}va_{y}=1. Thus a_{xvy} (t)=1=1v1=a_{x} (t)va_{y}(t).
The other case is that either xÎt or yÎt. Then either a_{x}(t)=o or
a_{y}(t)=0. And if xvyÎt then because the t is an ideal we
deduce that x,yÎt thus a_{xvy}(t)=0=0v0=a_{x}(t)va_{y}(t).
If xvyÎt, then either xÎt or yÎt (in the contrary xvyÎt which is a contradiction). Thus
either xÎt and yÎt or xÎt and yÎt. In both cases a_{xvy}(t)=1=1v0=0v1=a_{x}(t)va_{y}(t).
c) We shall prove that if x¹y then h(x)¹h(y).
If x¹y because the SpecL is separating
there is tÎSpecL such that xÎt and y¹t. Thus a_{x}(t)=0 and a_{y}(t)=1
hence a_{x}¹a_{y} or h(x)¹h(y). By the a), b), c) we conclude
that the h is a monomorphism of the lattices L and h(L)£ _{}. But in the category of lattices a monomorphism and
epimorphism is also an isomorphism
Q.E.D.
Corollary 21. (The extended Birkhoff-Stone
theorem).
A lattice is with separating
spectrum if and only if it is distributive.
Proof: By the Birkhoff-Stone theorem
we get that any distributive lattice is with separating spectrum. By the
extended Stone’s theorem 20 we get that any lattice with separating spectrum is
isomorphic with a sublattice of a Boolean lattice. But any sublattice of a
Boolean lattice is a distributive lattice.
Q.E.D.
Remark: We conclude that any
distributive lattice has a representation with sets: This seems that it is
already known (see [Sirkoski,R 1969] Appendix §3 first page of the paragraph).
Our proof does not nevertheless make use of any distributive assumption for the
lattice. With our technique it is proved something more: The inverse of the
Birkhoff-Stone theorem.