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Direction cones for the representation
- f tomonoids
Direction cones for the representation of tomonoids Thomas - - PowerPoint PPT Presentation
Direction cones for the representation of tomonoids Thomas - - PowerPoint PPT Presentation
Direction cones for the representation of tomonoids Thomas Vetterlein Department of Knowledge-Based Mathematical Systems, Johannes Kepler University (Linz) June 2013 Background ( Lotfi Zadeh ) Fuzzy logic deals with (is supposed to deal with)
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Background
(Lotfi Zadeh) Fuzzy logic deals with (is supposed to deal with) vague properties:
0.2 0.4 0.6 0.8 1 low
normal elevated
Fuzzy sets to model the result of a blood test
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Background
(Lotfi Zadeh) Fuzzy logic deals with (is supposed to deal with) vague properties:
0.2 0.4 0.6 0.8 1 low
normal elevated
Fuzzy sets to model the result of a blood test The collection of vague propositions gives rise (is supposed to give rise) to a residuated ℓ-monoid (L; ∧, ∨, ⊙, →, 0, 1) (Petr H´ ajek).
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Algebras for fuzzy logic
We frequently deal with certain residuated ℓ-monoids called MTL-algebras (Ll. Godo, F. Esteva):
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Algebras for fuzzy logic
We frequently deal with certain residuated ℓ-monoids called MTL-algebras (Ll. Godo, F. Esteva):
strength
- f propositions
(strong) conjunction implication truth, falsity
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The finite case
Theorem (A. Ciabattoni, G. Metcalfe, F. Montagna)
MTL-algebras form a variety, which is generated by its totally ordered finite members.
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The finite case
Theorem (A. Ciabattoni, G. Metcalfe, F. Montagna)
MTL-algebras form a variety, which is generated by its totally ordered finite members. One of the big issues of many-valued logics: How can totally ordered finite MTL-algebras be described?
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Tomonoids
(E. Gabovich, J. J. Madden et al., ...) We identify finite totally ordered MTL-algebras with “c.p.f. tomonoids”:
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Tomonoids
(E. Gabovich, J. J. Madden et al., ...) We identify finite totally ordered MTL-algebras with “c.p.f. tomonoids”:
Definition
(L; , +, 0) is a totally ordered monoid, or tomonoid, if: (T1) (L; +, 0) is a monoid; (T2) is a translation-invariant total order: a b implies a + c b + c and c + a c + b.
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Tomonoids
(E. Gabovich, J. J. Madden et al., ...) We identify finite totally ordered MTL-algebras with “c.p.f. tomonoids”:
Definition
(L; , +, 0) is a totally ordered monoid, or tomonoid, if: (T1) (L; +, 0) is a monoid; (T2) is a translation-invariant total order: a b implies a + c b + c and c + a c + b. A tomonoid is called commutative if + is commutative; positive if 0 is the bottom element. finitely generated if L is so as a monoid.
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Congruences and orders on free monoids
Consider the free monoid over n 1 elements, (Nn; +, 0).
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Congruences and orders on free monoids
Consider the free monoid over n 1 elements, (Nn; +, 0). Let be a translation-invariant, positive total order on Nn. Then (Nn; , +, 0) is a c.p.f. tomonoid.
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Congruences and orders on free monoids
Consider the free monoid over n 1 elements, (Nn; +, 0). Let be a translation-invariant, positive total order on Nn. Then (Nn; , +, 0) is a c.p.f. tomonoid.
Definition
A tomonoid that is a quotient of a tomonoid Nn is called formally integral.
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Congruences and orders on free monoids
Consider the free monoid over n 1 elements, (Nn; +, 0). Let be a translation-invariant, positive total order on Nn. Then (Nn; , +, 0) is a c.p.f. tomonoid.
Definition
A tomonoid that is a quotient of a tomonoid Nn is called formally integral. can be described by a positive cone on (Zn; +, 0), making Zn a totally ordered Abelian group.
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Congruences and orders on free monoids
Consider the free monoid over n 1 elements, (Nn; +, 0). Let be a translation-invariant, positive total order on Nn. Then (Nn; , +, 0) is a c.p.f. tomonoid.
Definition
A tomonoid that is a quotient of a tomonoid Nn is called formally integral. can be described by a positive cone on (Zn; +, 0), making Zn a totally ordered Abelian group. However: Not all tomonoids are formally integral.
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Preorders
Consider the free monoid over n elements, (Nn; +, 0). Let be a translation-invariant, positive total preorder on Nn.
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Preorders
Consider the free monoid over n elements, (Nn; +, 0). Let be a translation-invariant, positive total preorder on Nn. Then the symmetrisation ≈ of is a tomonoid congruence, and (Nn≈; , +, 0≈) is a c.p.f. tomonoid.
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Preorders
Consider the free monoid over n elements, (Nn; +, 0). Let be a translation-invariant, positive total preorder on Nn. Then the symmetrisation ≈ of is a tomonoid congruence, and (Nn≈; , +, 0≈) is a c.p.f. tomonoid.
Theorem
All c.p.f. tomonoids arise in this way.
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Preorders
Consider the free monoid over n elements, (Nn; +, 0). Let be a translation-invariant, positive total preorder on Nn. Then the symmetrisation ≈ of is a tomonoid congruence, and (Nn≈; , +, 0≈) is a c.p.f. tomonoid.
Theorem
All c.p.f. tomonoids arise in this way. Indeed, given a monoid epimorphism Nn → L, we can pull back the total order on L to Nn.
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Preorders
Consider the free monoid over n elements, (Nn; +, 0). Let be a translation-invariant, positive total preorder on Nn. Then the symmetrisation ≈ of is a tomonoid congruence, and (Nn≈; , +, 0≈) is a c.p.f. tomonoid.
Theorem
All c.p.f. tomonoids arise in this way. Indeed, given a monoid epimorphism Nn → L, we can pull back the total order on L to Nn. Question: Can we describe by means of something like a positive cone?
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Positive cone and direction cone
A translation-invariant, positive total order on Nn is called a monomial order.
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Positive cone and direction cone
A translation-invariant, positive total order on Nn is called a monomial order. The describing positive cone is C = {z ∈ Zn : a b for any a, b ∈ Nn such that z = b − a.}
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Positive cone and direction cone
A translation-invariant, positive total order on Nn is called a monomial order. The describing positive cone is C = {z ∈ Zn : a b for any a, b ∈ Nn such that z = b − a.}
Definition
Let be a translation-invariant, positive total preorder on Nn. Then we call a monomial preorder.
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Positive cone and direction cone
A translation-invariant, positive total order on Nn is called a monomial order. The describing positive cone is C = {z ∈ Zn : a b for any a, b ∈ Nn such that z = b − a.}
Definition
Let be a translation-invariant, positive total preorder on Nn. Then we call a monomial preorder. Moreover, the direction cone of is C = {z ∈ Zn : a b for any a, b ∈ Nn such that z = b − a}.
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Direction cones
Theorem
C ⊆ Zn is the direction cone of a monomial preorder iff: (C1) Let z ∈ Nn. Then z ∈ C and, if z = 0, −z / ∈ C. (C2) Let (x1, . . . , xk), k 2, be an addable k-tuple of elements of C. Then x1 + . . . + xk ∈ C. (C3) For each z ∈ Zn, either z ∈ C or −z ∈ C.
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Direction cones
Theorem
C ⊆ Zn is the direction cone of a monomial preorder iff: (C1) Let z ∈ Nn. Then z ∈ C and, if z = 0, −z / ∈ C. (C2) Let (x1, . . . , xk), k 2, be an addable k-tuple of elements of C. Then x1 + . . . + xk ∈ C. (C3) For each z ∈ Zn, either z ∈ C or −z ∈ C. (x1, . . . , xk) is addable if for i = 1, . . . , k xi + . . . + xk (x1 + . . . + xk) ∨ 0.
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Cone tomonoids
A monomial preorder has a direction cone C.
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Cone tomonoids
A monomial preorder has a direction cone C. A direction cone defines in turn a monomial preorder:
Definition
Let C ⊆ Zn be a direction cone. Then the monomial preorder induced by C is the smallest preorder C such that (O) a C b for any a, b ∈ Nn such that b − a ∈ C.
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Cone tomonoids
A monomial preorder has a direction cone C. A direction cone defines in turn a monomial preorder:
Definition
Let C ⊆ Zn be a direction cone. Then the monomial preorder induced by C is the smallest preorder C such that (O) a C b for any a, b ∈ Nn such that b − a ∈ C. The tomonoid represented by C is called a cone tomonoid.
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Cone tomonoids
A monomial preorder has a direction cone C. A direction cone defines in turn a monomial preorder:
Definition
Let C ⊆ Zn be a direction cone. Then the monomial preorder induced by C is the smallest preorder C such that (O) a C b for any a, b ∈ Nn such that b − a ∈ C. The tomonoid represented by C is called a cone tomonoid. C is contained in , hence:
Theorem
Any c.p.f. tomonoid is the quotient of a cone tomonoid.
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Example
Let L be a tomonoid generated by a and b: 0 < a < b < 2a < a + b < 2b < 3a < 2a + b < a + 2b = 4a < 1.
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Example
Let L be a tomonoid generated by a and b: 0 < a < b < 2a < a + b < 2b < 3a < 2a + b < a + 2b = 4a < 1. The monomial preorder representing L.
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Example, ctd.
The direction cone of .
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Example, ctd.
The cone tomonoid whose quotient is L.
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Summary so far
◮ Any c.p.f. tomonoid is a quotient of a cone tomonoid.
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Summary so far
◮ Any c.p.f. tomonoid is a quotient of a cone tomonoid. ◮ A cone tomonoid is specified by a direction cone,
which is a subset of a Zn subject to conditions similar to the case of positive group cones.
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The finite case
Let (L; , +, 0) be finite.
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The finite case
Let (L; , +, 0) be finite. Drawback: The direction cone describing L is infinite (unless L is trivial).
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The finite case
Let (L; , +, 0) be finite. Drawback: The direction cone describing L is infinite (unless L is trivial). Solution: Let ≈ be the congruence on Nn inducing the finite tomonoid L.
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The finite case
Let (L; , +, 0) be finite. Drawback: The direction cone describing L is infinite (unless L is trivial). Solution: Let ≈ be the congruence on Nn inducing the finite tomonoid L. Then we choose S (“support”), a finite subset of Nn having a non-empty intersection with each ≈-class.
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The finite case
Let (L; , +, 0) be finite. Drawback: The direction cone describing L is infinite (unless L is trivial). Solution: Let ≈ be the congruence on Nn inducing the finite tomonoid L. Then we choose S (“support”), a finite subset of Nn having a non-empty intersection with each ≈-class. We include into the direction cone only differences of elements of S.
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Example, again
The support of .
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Example, ctd.
The direction f-cone of .
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Summary
◮ Any finite c.p.f. tomonoid is a quotient
- f an f-cone tomonoid.
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Summary
◮ Any finite c.p.f. tomonoid is a quotient
- f an f-cone tomonoid.
◮ An f-cone tomonoid is specified by the pair (S, C), where
S, the support, is a finite -ideal of Nn; C, the f-cone, is a subset of the set of differences of elements of S.
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Summary
◮ Any finite c.p.f. tomonoid is a quotient
- f an f-cone tomonoid.