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Fuzzy sets as two-sorted algebras
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Fuzzy sets as two-sorted algebras Ulrich Höhle Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Wuppertal, Germany Received 10 October 2014; received in revised form 3 April 2015; accepted 6 May 2015
Abstract This paper makes the attempt to explain the foundations of fuzzy sets from the point of view of universal algebra. The free fuzzy set and the fuzzy power set are constructed. Moreover, fuzzy power sets give rise to a monad. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets. © 2015 Elsevier B.V. All rights reserved. Keywords: Frobenius -monoid; 2-Sorted algebra; Fuzzy set; Fuzzy power set; Fuzzy module
1. Introduction and motivation The fiftieth anniversary of fuzzy sets is a good occasion to go back to their roots and to explain the impact of universal algebra on fuzzy set theory. Therefore I do not consider fuzzy sets as generalized characteristic functions (cf. [7]) or as enriched presheaves (cf. [9]), but simply as two-sorted algebras without any further specific meaning. Before I put down the axioms of this type of algebra it seems to be interesting to consult first the well known paper on fuzzy sets [18] published by L.A. Zadeh in 1965, even though the principal idea of a fuzzy set goes back to K. Menger (cf. [14]). fA In Section 2 of [18] a fuzzy set (class) A in X is not defined, but characterized by a function X −−− → [0, 1] which L.A. Zadeh understands as membership (characteristic) function assigning a real number of the interval [0, 1] to each point in X. Also S. Mac Lane in his critical review on the health of mathematics admits that this idea is attractive and gives the following explanation of this function (cf. p. 54 in [13]): . . . instead of saying that an element x is or is not in the set A, let us measure the likelihood that x is in A.
As I said before I do not continue the debate on membership and measurement of membership, but I understand simply a fuzzy set in X as map from X to [0, 1]. The interesting question is now which algebraic properties of [0, 1] have been used by L.A. Zadeh for his calculus of fuzzy sets. A short glance at [18] shows that he has selected four operations E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2015.05.004 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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– one unary operation sending each element α ∈ [0, 1] to 1 − α, – three binary operations given by the minimum, maximum and product, and a partially defined operation determined by the addition which he calls algebraic sum (cf. p. 344 in [18]). For the contemporary it is unclear why L.A. Zadeh did not complete the algebraic sum to a totally defined binary operation which would coincide with the t -conorm corresponding to Łukasiewicz arithmetic conjunction determined by: α ∗ β = max(α + β − 1, 0),
α, β ∈ [0, 1].
But in a more mathematical language it is clear that L.A. Zadeh has used the standard structure of an MV-algebra on [0, 1] (cf. [4,8]) ([0, 1], min, max, ∗) in which the product · plays the role of a dominating binary operation (cf. [17]) — i.e. (α · β) ∗ (γ · δ) ≤ (α ∗ γ ) · (β ∗ δ),
α, β, γ , δ ∈ [0, 1].
Since MV-algebras are divisible and commutative Frobenius -monoids (cf. [1]), I put down the following abstract definition of a fuzzy set: f
A triple (X, f, Q) is called a fuzzy set iff X is a set, Q is a Frobenius -monoid and X −→ Q is a map.
The aim of this paper is to show that fuzzy sets in the previous sense are two-sorted algebras — i.e. the underlying Frobenius -monoid is incorporated into the definition of a fuzzy set and the membership map f is viewed as a unary two-sorted operation (cf. [6,2]).1 The precise definition of the corresponding two-sorted signature will be given in Section 4. Proceeding in this way the techniques of universal algebra are available, and it is not surprising that free fuzzy sets exist. In particular, every fuzzy set is a quotient of a free fuzzy set. Moreover, in the case of Frobenius quantales the fuzzy power set construction and its monadic basis are explained. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets. As a preparation I first explain some basic properties of Frobenius -monoids. 2. Frobenius -monoids A residuated -semigroup is a quintuple (X, ≤, ∗, , ) where (X, ∗) is a semigroup, (X, ≤) is a lattice with the corresponding binary lattice operations ∧ and ∨, and and are two further binary operations on X satisfying the following condition for all x, y, z ∈ X (cf. [1]): y ≤ x z ⇔ x ∗ y ≤ z ⇔ x ≤ z y.
(AD)
In this context is called the right-implication and is called the left-implication w.r.t. the semigroup operation ∗. The next proposition collects some fundamental properties of residuated -semigroups. Proposition 2.1. Let ((X, ≤), ∗, , ) be residuated -semigroup. Then the semigroup operation ∗ is distributive on binary joins in each variable separately — i.e. x ∗ (y ∨ z) = (x ∗ y) ∨ (x ∗ z)
and (x ∨ y) ∗ z = (x ∗ z) ∨ (y ∗ z).
(D)
If ⊥ is the universal lower bound of (X, ≤), then ⊥ is always the zero element of (X, ∗). Moreover the following properties hold for all x, y, z ∈ X: (i) (x ∨ y) z = (x z) ∧ (y z) and z (x ∨ y) = (z x) ∧ (z y). (ii) x (y ∧ z) = (x y) ∧ (x z) and (z ∧ y) x = (z x) ∧ (y x). 1 It is interesting to see that the term many-sorted has its origin in mathematical logic (cf. [16]) and plays now a prominent role in certain areas of algebra (cf. [15]).
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(iii) (x ∗ y) z = y (x z) and z (y ∗ x) = (z x) y. (iv) (x y) z = x (y z). Even though the previous proposition is well known (see e.g. Theorem 4 on p. 326 in [1]), I give here a direct proof of Proposition 2.1 for the convenience of the reader. Proof. Because of the reflexivity and transitivity axiom of partial orders the isotonicity of the semigroup operation ∗ is an immediate corollary of condition (AD). Hence with regard to (D) I have only the verify the inequalities x ∗ (y ∨ z) ≤ (x ∗ y) ∨ (x ∗ z)
and
(x ∨ y) ∗ z ≤ (x ∗ z) ∨ (y ∗ z).
For this purpose I put u = (x ∗ y) ∨ (x ∗ z) and infer from (AD) that y ∨ z ≤ x u holds. Now I apply again (AD) and obtain x ∗ (y ∨ z) ≤ u. Replacing the right-implication by the left-implication the verification of (x ∨ y) ∗ z ≤ (x ∗ z) ∨ (y ∗ z) follows by means of an analogous argument. Finally, I assume that the universal lower bound ⊥ exists in (X, ≤), and fix an arbitrary element x ∈ X. Since ⊥ ≤ x z holds for all z ∈ X, I conclude form (AD) that x ∗ ⊥ coincides with the universal lower bound. Analogously I verify ⊥ ∗ x = ⊥. Hence ⊥ is the zero element of (X, ∗). Now let us turn to the properties (i)–(iv). Because of the isotonicity of ∗ it is easily seen that the inequalities (x ∨ y) z ≤ (x z) ∧ (y z) and z (x ∨ y) ≤ (z x) ∧ (z y) hold. Since ∗ is distributive on binary joins, one infers from (AD) that also the converse inequalities are valid. Hence (i) is verified. The relation (ii) is an immediate corollary from the isotonicity of ∗. Finally, (iii) and (iv) follow from the associativity of ∗. In fact, the following chain of equivalences holds true: v ≤ (x ∗ y) z ⇔ x ∗ y ∗ v ≤ z ⇔ y ∗ v ≤ x z ⇔ v ≤ y (x z). Analogously the relation z (y ∗ x) = (z x) y can be verified. The subsequent chain of equivalences v ≤ (x y) z ⇔ v ∗ z ≤ x y ⇔ x ∗ v ∗ y ≤ z ⇔ x ∗ v ≤ z y ⇔ v ≤ x (z y) ensures the validity of (v).
2
An element d of a residuated -semigroup (X, ≤, ∗, , ) is called dualizing, if for all x ∈ X the following relations hold: d (x d) = x
and
(d x) d = x.
(2.1)
Proposition 2.2. Let (X, ≤, ∗, , ) be a residuated -semigroup with a dualizing element d. Then the following properties hold: (i) d d is a right-unit and d d is a left-unit w.r.t. ∗ — this means x ∗ (d d) = x and (d d) ∗ x = x for all x ∈ X. (ii) d d and d d coincide and form the unit of (Q, ∗). Moreover, if (X, ≤) be a bounded lattice with universal bounds ⊥ and , then the additional properties are valid (iii) d = ⊥ = d and ⊥ d = = d ⊥. Proof. In order to show that d d is a right-unit of (X, ∗) I fix x ∈ X and use the property (iii) in Proposition 2.1 and the dualizing property of d:
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x ∗ (d d) = d (x ∗ (d d)) d = d (d d) x d = (d x) d = x. Hence d d is a right-unit in (X, ∗). Analogously we verify that (d d) is a left-unit in (Q, ∗). Finally, (ii) follows immediately from (i). Now we assume that (X, ≤) has universal bounds. Since ⊥ is the zero element of (X, ∗) (cf. Proposition 2.1) the relations ⊥ d = and d ⊥ = hold. Hence (iii) follows from (2.1). 2 The previous proposition shows that every residuated -semigroup with a dualizing element is necessarily unital. This observation is a motivation to introduce the following terminology. Definition 2.1. A sextuple (X, ≤, ∗, , , d) is called a Frobenius -monoid, if (X, ≤, ∗, , ) is a residuated -semigroup and d is a dualizing element in (X, ≤, ∗, , ). The next proposition explains first properties of Frobenius -monoids. Proposition 2.3. In every Frobenius -monoid (X, ≤, ∗, , , d) the following properties hold for all x, z ∈ X: (i) (ii) (iii) (iv)
x z = ((d z) ∗ x) d. z x = d (x ∗ (z d)). (x ∧ z) d = (x d) ∨ (z d). d (x ∧ z) = (d x) ∨ (d z).
Proof. Because of (2.1) and the property (iii) in Proposition 2.1 the relation x z = x ((d z) d) = ((d z) ∗ x) d holds Hence (i) is verified. Further, (iii) follows from the following relation: (x ∧ z) d = d (x d) ∨ (z d) d = (x d) ∨ (z d). Finally, (ii) and (iv) can be verified analogously. 2 Left-sided and right-sided elements play a special role in Frobenius -monoids. For this purpose I recall the following terminology (cf. [1]). Definition 2.2. Let (X, ≤, ∗, , ) be a residuated -semigroup such that the universal upper bound exists in (X, ≤). An element x ∈ X is called left-sided (right-sided) if ∗ x ≤ x (x ∗ ≤ x) holds. An element x of X is said to be two-sided if x is left-sided and right-sided. Lemma 2.4. Let (X, ≤, ∗, , , d) be a Frobenius -monoid such that (X, ≤) is a bounded lattice. Then the following properties hold: (i) If x ∈ X is left-sided, then x d = x ⊥ holds. (ii) If x ∈ X is right-sided, then d x = ⊥ x holds. Proof. In order to verify (i) we proceed as follows. Let z be a left-sided element of X with z ≤ d. Then d z = holds. Hence z = ⊥ follows from Proposition 2.2(iii). Now we choose an arbitrary left-sided element x. Since ∗ is associative, x ∗ (x d) is left-sided, and we deduce from x ∗ (x d) ≤ d that x ∗ (x d) = ⊥ is valid. Hence x d = x ⊥ follows. — Analogously we prove (ii). 2
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Corollary 2.5. Every Frobenius -monoid with universal bounds satisfies the Frobenius property — i.e. for all leftsided elements x and for all right-sided elements y the following relation holds: x = ⊥ (x ⊥) and
y = (⊥ y) ⊥.
(Frobenius Property)
Proof. Let (X, ≤, ∗, , , d) be a Frobenius -monoid with universal bounds and let x ∈ X be right-sided. Then the relation d x = ⊥ x follows from Lemma 2.4(ii). Since ⊥ is the zero element of (X, ∗) (cf. Proposition 2.1), the element ⊥ x is left-sided. Now I use Lemma 2.4(i) and obtain: x = (d x) d = (⊥ x) d = (⊥ x) ⊥. If y is left-sided, then the property y = ⊥ (y ⊥) can be verified analogously. 2 Let (X, ≤, ∗, , , d) and (X, ≤, ∗, , , d) be Frobenius -monoids. A map X −−h−→ Y is called a Frobenius morphism, if h is a residuated -semigroup homomorphism and preserves the respective dualizing elements — i.e. h(x ∧ y) = h(x) ∧ h(y),
h(x y) = h(x) h(y),
h(x ∗ y) = h(x) ∗ h(y),
h(x ∨ y) = h(x) ∨ h(y),
h(y x) = h(y) h(x),
h(d) = d.
Since the unit in any Frobenius -monoid with dualizing element d has the form d d (cf. Proposition 2.2(ii)), it follows that Frobenius morphisms are always unital. Because of Proposition 2.3 Frobenius morphisms can be characterized as follows. Proposition 2.6. Let (X, ≤, ∗, , , d) and (Y, ≤, ∗, , , d) be Frobenius -monoids, and let map X −−h−→ Y be a map. Then the following assertions are equivalent: (i) h is a Frobenius morphism. (ii) h preserves the respective dualizing elements and satisfies the properties h(x ∗ z) = h(x) ∗ h(z), h(x d) = h(x) d, h(d z) = d h(z), h(x ∨ z) = h(x) ∨ h(z) for all x, z ∈ X. Proof. The implication (i) ⇒ (ii) is evident. On the other hand, if (ii) holds, then we infer form Proposition 2.3(i) and Proposition 2.3(ii) that h preserves the left- and right-implication. Finally, we use Proposition 2.3(iv) and Proposition 2.1(i) and obtain: h(x ∧ z) = h(d (x ∧ z)) d = h((d x) ∨ (d z)) d = h(d x) ∨ h(d z) d = (d h(x)) d ∧ (d h(z)) d = h(x) ∧ h(z). Hence h also preserves the binary meet operation. 2 The class of Frobenius -monoids and Frobenius morphisms form a category Frob in an obvious way. 3. Frobenius -monoids as algebras It is well known (cf. [3]) that every lattice (X, ≤) forms an algebra (X, , ⊕) where and ⊕ are commutative and associative, binary operations on X satisfying the absorption laws: x (x ⊕ y) = x
and
x ⊕ (x y) = x
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for all x, y ∈ X. In this context plays the role of binary meets and ⊕ the role of binary joins. Further, any semigroup is an algebra with a unique associative binary operation. Hence in order to understand Frobenius -monoids as algebras it is only necessary to formulate an equational characterization of condition (AD). Theorem 3.1. Let (X, ≤) be a lattice provided with an associative and binary operations ∗. Further, let and two further binary operations on X. Then the following assertions are equivalent: (i) The quintuple (X, ≤, ∗, , ) is a residuated -semigroup. (ii) The following properties hold: (a) z = z ∨ (x ∗ ((x z) ∧ y)), y = y ∧ (x ((x ∗ y) ∨ z)), x, y, z ∈ X. (b) z = z ∨ (((z x) ∧ y) ∗ x), y = y ∧ (((y ∗ x) ∨ z) x), x, y, z ∈ X. Proof. If (i) holds, then the properties (a) and (b) follow from the isotonicity of the semigroup operation ∗ (cf. (D)), the property (ii) in Proposition 2.1 and the following observations: x ∗ (x z) ≤ z,
y ≤ x (x ∗ y),
(z x) ∗ x ≤ z,
y ≤ (y ∗ x) x).
On the other hand, the property (a) implies the equivalence x ∗ y ≤ z ⇔ y ≤ x z. In fact, the relation x ∗ y ≤ z implies y = y ∧ (x z) — i.e. y ≤ x z, and the relation y ≤ x z implies z = z ∨ (x ∗ y) — i.e. x ∗ y ≤ z. Analogously, it can be shown that the property (b) implies the equivalence y ∗ x ≤ z ⇔ y ≤ z x. Hence (ii) implies the condition (AD). 2 Referring to (2.1) a Frobenius -monoid is an algebra of the following form (X, , ⊕, ∗, , , d) where d is a nullary operation (i.e. a constant) of X and , ⊕, ∗, , are binary operations on X satisfying the following axioms: (1) (2) (3) (4) (5) (6)
, ⊕ and ∗ are associative operations. and ⊕ are commutative operations. x (x ⊕ y) = x and x ⊕ (x y) = x. z = z ⊕ (x ∗ ((x z) y)), y = y (x ((x ∗ y) ⊕ z)). z = z ⊕ (((z x) y) ∗ x), y = y (((y ∗ x) ⊕ z) x). x = d (x d), and x = (d x) d.
(Absorption Laws)
Hence the (one-sorted) signature = (n )n∈N0 of a Frobenius -monoid has the form: 0 = {ω0 }, 1 = ∅, 2 = {ω1 , ω2 , ω3 , ω4 , ω5 }, n = ∅ for 3 ≤ n. Then we conclude from the (one-sorted) term construction in Set (cf. [3,5]) that for every set X the free -algebra X exists and is given by the set of all -terms in X. In order to arrive at the free Frobenius -monoid generated by X it is necessary to form an appropriate quotient of the free -algebra X fulfilling the equation in (1)–(6). Hence the subsequent theorem holds. Theorem 3.2. Let X be a set. Then there exists a Frobenius -monoid (X , ≤ , ∗ , , , d )
and a map
η
X X −−− → X
such that for any further Frobenius -monoid (Y, ≤, ∗, , , d) and for any further map X −−h−→ X there exists a
unique Frobenius morphism X −−h−→ Y making the following diagram commutative: X
ηX
X
h
h
Y
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Theorem 3.2 can also expressed as the statement that the forgetful functor from Frob to Set has a left adjoint. In this sense the choice of morphisms between Frobenius -monoids was not free in Section 2, but dictated by methods of universal algebra. 4. Fuzzy sets as two-sorted algebras Now I take up the abstract definition of a fuzzy set given in Section 1 and recall that a fuzzy set is a triple (X, f, Q) f where X is a set, Q is a Frobenius -monoid, and X −→ Q is a map. Since Frobenius monoids are algebras (cf. Section 3), it is evident that fuzzy sets are two-sorted algebras. Hence a morphism from a fuzzy set (X, f, Q) to a ϕ h R is a Frobenius morphism fuzzy set (Y, g, R) is a pair of maps (ϕ, h) where X −→ Y is a simple map and Q −→ such that the following diagram is commutative in Set: ϕ
X
Y g
f
Q
h
R
The category of fuzzy sets and their morphisms is denoted by FSet. In the following consideration I specify the two-sorted signature of a fuzzy set. For this purpose I recall the free monoid M generated by {1, 2} — i.e. M = {(0, ∅)} ∪
{n} × {1, 2}n .
n∈N (σ,s)
Then a two-signature = (, {1, 2}) is a system (n )(n,σ,s)∈M×{1,2} of sets indexed by M × {1, 2}. Elements (σ,s) are also called two-sorted functional symbols of arity n. Because of this semantic meaning the following of n (σ,s) notation for n is used in computer science: (∅,s)
0
= →s 0 ,
(σ,s) = sn1 ×...×sn →s , n
1 ≤ n, σ = (s1 , . . . , sn ).
After these preparations I specify the two-sorted signature F = (, {1, 2}) of a fuzzy set: →1 0 = ∅, →2 0 = {ω0 }, 1→2 = {ω1 2 }, 1 2×2→2 = {ω1 , ω2 , ω3 , ω4 , ω5 }, 2 sn1 ×...×sn →s = ∅,
elsewhere.
Then the two-sorted term construction in Set shows that for any pair (X1 , X2 ) of sets X1 and X2 the free -algebra of F -terms in (X1 , X2 ) exists (see e.g. [5]). Subsequently, an appropriate quotient of the free F -algebra leads to the free fuzzy set generated by (X1 , X2 ). In the following consideration I give a direct proof of this standard fact. Theorem 4.1. Let (X1 , X2 ) be a pair of sets and X1 X2 be the disjoint join of X1 and X2 with the canonical jX
jX
1 2 embeddings X1 −−→ X1 X2 and X2 −−→ X1 X2 . Further, let Q be the free Frobenius -monoid generated by ηX1 X2 X1 X2 with the embedding X1 X2 −−−−−→ Q. Then (X1 , π, Q) with π = ηX1 X2 ◦ jX1 is a fuzzy set. Moreover, for ϕ h any further fuzzy set (Y, g, R) and for any further pair (ϕ, h) of maps X1 −→ Y and X2 −→ R there exists a unique (ψ,k) fuzzy set morphism (X, π, Q) −−−−→ (Y, g, R) making the following diagram commutative in Set × Set:
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(X1 , X2 )
(1X1 ,ηX X2 ◦jX2 )
(X1 , Q)
(ψ,k)
(ϕ,h)
(E)
(Y, R) — i.e. ψ = ϕ and h = k ◦ (ηX X2 ◦ jX2 ). Proof. (a) (Uniqueness) Let us assume that (ψ, k) is a fuzzy set morphism making the diagram (E) commutative. Hence ψ =ϕ
and k ◦ (ηX X2 ◦ jX2 ) = h
(4.1)
follow. Since (ψ, k) is a fuzzy set morphism from (X1 , π, Q) to (Y, g, R), the relation k ◦ ηX1 X2 ◦ jX1 = k ◦ π = g ◦ ϕ
(4.2)
holds. Let us recall that the coproduct in Set is given by disjoint unions. Hence there exists a unique map X 1 X2
(g◦ϕ)h
R satisfying the properties (g ◦ ϕ) h) ◦ jX1 = g ◦ ϕ and (g ◦ ϕ) h) ◦ jX2 = h. In particular, the relation k ◦ ηX1 X2 = (g ◦ ϕ) h follows from (4.1) and (4.2). Since k is a Frobenius morphism, I conclude from Theorem 3.2 that k coincides with the unique extension of (g ◦ ϕ) h to a Frobenius morphism from Q to R. Hence the uniqueness of (ψ, k) is verified. (b) (Existence) Let (g ◦ ϕ) h be the unique extension of (g ◦ ϕ) h to a Frobenius morphism from Q to R (cf. Theorem 3.2). Then the argumentation in (a) suggests to define (ψ, k) as follows: ψ = ϕ and k = (g ◦ ϕ) h . We have to show that (ψ, k) is a fuzzy set morphism making the diagram (E) commutative. First it is easily seen that the following relation holds: k ◦ π = (g ◦ ϕ) h ◦ ηX1 X2 ◦ jX1 = (g ◦ ϕ) h ◦ jX1 = g ◦ ϕ = g ◦ ψ. Hence (ψ, k) is a fuzzy set morphism. A further observation is the following one: k ◦ (ηX1 X2 ◦ jX2 ) = (g ◦ ϕ) h ◦ ηX1 X2 ◦ jX2 = (g ◦ ϕ) h ◦ jX2 = h. Hence the diagram (E) is commutative. 2 The previous theorem can also be expressed by the statement that the forgetful functor from the category FSet of fuzzy sets to Set × Set has a left adjoint. In particular, in the sense of FSet every fuzzy set is a quotient of an appropriate free fuzzy set. 5. Fuzzy power set monad In this section I assume that the underlying lattice of some given Frobenius -monoid is complete — this means that Q is a Frobenius quantale. In this context I recall that the left-implication and right-implication of a Frobenius quantale have the following form:
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β α =
{γ ∈ Q | γ ∗ α ≤ β} and α β =
9
{γ ∈ Q | α ∗ γ ≤ β}.
Further, every Frobenius quantale Q with the dualizing element d satisfies the following properties A d = {a d | a ∈ A} and d A = {d a | a ∈ A}
(5.1)
for all subsets A of Q. In fact, if A is empty, then (5.1) follows from Proposition 2.2(iii). In the case of non-empty subsets the proof of Proposition 2.3(iii) and (iv) carries over. Morphisms between Frobenius quantales are Frobenius morphisms which are join preserving. Referring to (5.1) it is easily seen that join preserving Frobenius morphisms are always meet preserving. Moreover, the MacNeille completion of every Frobenius -monoid is a Frobenius quantale (for the commutative case see [8]), and every Frobenius morphism having a right adjoint has a unique extension to a join preserving Frobenius morphism between the corresponding Frobenius quantales. Let FQSet be the subcategory of FSet consisting of fuzzy sets (X, f, Q) and fuzzy set morphisms (ϕ, h) s.t. Q is a Frobenius quantale and h is a join preserving. The aim of this section is to construct a monad on FQSet which can be understood as the fuzzy power set monad. First we define an object function P: |FQSet| → |FQSet| as follows. With every fuzzy set (X, f, Q) we associate a fuzzy set P(X, f, Q) := (QX , F, Q) where the map QX −F→ Q is determined by: F (g) = f (x) g(x), g ∈ QX . (5.2) x∈X
The fuzzy set P(X, f, Q) is also called the fuzzy power set of (X, f, Q). Now I complete the object function P (ϕ,h) to an endofunctor of FQSet as follows. For every fuzzy set morphism (X, f, Q) −−−→ (Y, g, R) I define a pair P(ϕ, h) := (P (ϕ), h) of maps by P (ϕ)(m)(y) = {h ◦ m(x) | ϕ(x) = y}, m ∈ QX , y ∈ Y (5.3) and show that (P (ϕ), h) is a fuzzy set morphism from P(X, f, Q) to P(Y, g, R). For this purpose I fix m ∈ QX . Since h is meet preserving, I obtain: h ◦ F (m) = h(f (x)) h(m(x)) x∈X
=
g(ϕ(x)) h(m(x))
x∈X
=
g(y)
y∈Y
=
{h(m(x))|ϕ(x) = y}
g(y) P (ϕ)(m)(y)
y∈Y
= G(P (ϕ)(m)). Hence P(ϕ, h) is a fuzzy set morphism. It is easily seen that P preserves the composition in FQSet. Finally we introduce two natural transformations η: idFQSet → P and μ: P ◦ P → P. For this purpose the unit (resp. universal lower bound) of the respective Frobenius quantales is denoted by e (resp. ⊥): η(X,f,Q) =(jX ,1Q )
(X, f, Q) jX (x) = e · 1x Because of f (x) =
P(X, f, Q) e, z = x and e · 1x (z) = ⊥, z = x,
z∈X f (z) e · 1x (z) = F (e · 1x )
where x, z ∈ X. it is evident that (jX , 1Q ) is a fuzzy set morphism. Further the (ϕ,h)
relation P (ϕ)(e · 1x ) = h(e) · 1ϕ(x) holds for every fuzzy set morphism (X, f, Q) −−−→ (Y, g, R) where I recall that h(e) is the unit in R. Hence η with components η(X,f,Q) is a natural transformation from idFQSet to P.
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The second natural transformation μ with components μ(X,f,Q) =(νX ,1Q )
P(P(X, f, Q)) has the form: νX (K)(x) =
P(X, f, Q)
X
K(k) ∗ k(x),
K ∈ Q(Q ) .
k∈QX
Because of f (x) νX (K)(x) = f (x) K(k) ∗ k(x) x∈X
x∈X
k∈QX
=
f (x) k(x) K(k)
x∈X
k∈QX
X
the pair (νX , 1Q ) is a fuzzy set morphism. Moreover for all K ∈ Q(Q ) and y ∈ Y the following relation holds: [νY (P (P (ϕ))(K))](y) = {h ◦ K(k) | P (ϕ)(k) = ∗ (y) ∈QY
=
h ◦ K(k) ∗
{h ◦ k(x) | ϕ(x) = y}
k∈QX
=
{h K(k) ∗ k(x)) | ϕ(x) = y}
k∈QX
= [P (ϕ)(νX (K))](y). Hence μ with components μ(X,f,Q) is a natural transformation from P ◦ P to P. Theorem 5.1. The triple (P, η, μ) is a monad on FQSet. Proof. Let (X, f, Q) be a fuzzy set. Referring to [12] I have to show the commutativity of the following diagrams: P(X, f, Q)
ηP(X,f,Q)
P(P(X, f, Q))
P (η(X,f,Q) )
μ(X,f,Q)
1P(X,f,Q)
P(X, f, Q)
1P(X,f,Q)
(Unit)
P(X, f, Q) P(P(P(X, f, Q)))
P (μ(X,f,Q) )
μP(X,f,Q)
P(P(X, f, Q))
P(P(X, f, Q))
μ(X,f,Q)
μ(X,f,Q)
(Associativity)
P(X, f, Q)
Let k be an element of QX . It is easily seen that the subsequent relations hold: νX (e · 1k ) = k, [P (jX )(k)]() = {k(x) | e · 1x = }, νX (P (jX )(k)) = k. Hence the unit axiom is verified. Since for K ∈ Q(Q
QX )
and x ∈ X one obtains:
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U. Höhle / Fuzzy Sets and Systems ••• (••••) •••–•••
K(K) ∗ K(k) ∗ k(x) =
k∈QX , K∈Q
QX
K∈Q
11
K(K) ∗ νX (K)(x)
QX
= νX (P (νX )(K))(x), the associativity axiom holds also true. 2 The previous theorem shows that a non-trivial “power set theory” is available for fuzzy sets viewed as two-sorted algebras. This observation is a motivation to call the monad in Theorem 5.1 the fuzzy power set monad. 6. Fuzzy modules In this section I show that the fuzzy power set is the free fuzzy module generated by its underlying fuzzy set. First I need some terminology. Definition 6.1. A left module on a Frobenius quantale is a triple (X, Q, ) where X is a complete lattice, Q is a Frobenius quantale and is a left action on X w.r.t. Q in the sense of the category Sup of all complete lattices and ⊗ join preserving maps — this means that Q × X −→ X is an outer binary operation on X satisfying the following axioms (cf. [11,10]):
(i) The map Q × X −−−→ X is join preserving in each variable separately — i.e. αi x = αi x and α ( xi = α xi . i∈I
i∈I
i∈I
i∈I
(ii) α (β x) = (α ∗ β) x where ∗ denotes the multiplication in Q. (iii) e x = x where e is the unit in Q. Let (X, Q, ) and (Y, R, ) be left modules on Frobenius quantales. A module homomorphism from (X, Q, ) to ϕ h R s.t. ϕ is join preserving, h is a join preserving Frobenius morphism, (Y, R, ) is a pair of maps X −→ Y and Q −→ and the following diagram is commutative: Q×X
h×ϕ
R×Y
X
ϕ
Y
Left modules on Frobenius quantales and morphism between them form a category in an obvious way. From the point of view of many valued mathematics the interesting point is here that any left module (X, Q, ) has an intrinsic Q-valued preorder pX determined by: pX (x2 , x1 ) = {α ∈ Q | α x1 ≤ x2 }, x1 , x2 ∈ X. (6.1) This observation motives the following definition. Definition 6.2. A quintuple (X, x0 , f, Q, ) is a fuzzy left module if x0 is an element of X, (X, f, Q) is a fuzzy set and (X, Q, ) is a left module such that the property f (x) = {α ∈ Q | α x ≤ x0 } holds for all x ∈ X — i.e. f (x) coincides with the value pX (x0 , x) of the intrinsic Q-valued preorder at (x0 , x) for all x ∈ X (cf. (6.1)). A pair (ϕ, h) is a fuzzy module homomorphism iff (ϕ, h) is a fuzzy set morphism and a module homomorphism. It is easily seen that fuzzy modules and fuzzy module homomorphisms form a category FMod. The next example explains that every fuzzy power set gives rise to a fuzzy module.
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Example 6.1. Let (X, f, Q) be a fuzzy set and P(X, f, Q) = (QX , F, Q) be its fuzzy power set. Further, let ∗ be the multiplication in Q and be the left action on QX defined by (α g)(x) = α ∗ g(x),
α ∈ Q,
x ∈ X,
g ∈ QX .
(6.2)
Then (QX , Q, ) is a left module. Obviously, the intrinsic Q-valued preorder p on QX has the form (cf. (6.1)): p(k, g) = {α ∈ Q | α g ≤ k} = k(x) g(x), g, k ∈ QX . x∈X
Hence we conclude from the definition of F (cf. (5.2)) that the fuzzy power set P(X, f, Q) can be enriched to a fuzzy left module as follows: (QX , f, F, Q, ). In this sense every fuzzy power set can be understood as a fuzzy left module where is given by (6.2). Now we prove the main theorem of this section which says that the fuzzy power set is the free fuzzy module generated by its underlying fuzzy set. (ϕ,h)
Theorem 6.1. Let (X, f, Q) be a fuzzy set and (Y, y0 , g, R, ) be a fuzzy left module. Further, let (X, f, Q) −−−→ (ϕ ,h )
(Y, g, R) be a fuzzy morphism. Then there exists a unique fuzzy module homomorphism (QX , f, F, Q, ) −−−−−→ (Y, y0 , g, R, ) such that the following diagram is commutative in FQSet: (X, f, Q)
η(X,f,Q)
P(X, f, Q) (D)
(ϕ ,h )
(ϕ,h)
(Y, g, R) In particular, h and h coincide. Proof. Let us begin with some general observations. Since the second component of η(X,f,Q) is the identity of Q, the join preserving Frobenius morphisms h and h coincide. Moreover, every element a ∈ QX can be represented as follows: a= a(x) jX (x) (6.3) x∈X
where is determined by (6.2) and jX has the form (cf. Section 5): jX (x) = e · 1x ,
e is the unit of Q.
(a) (Uniqueness) Let (ϑ, k) be a fuzzy module homomorphism making the diagram (D) commutative. Hence h and k coincide. In order to show that ϑ is uniquely determined, we use the fact that (ϑ, h) is a fortiori a module homomorphism and derive the following relation form (6.3): ϑ(a) = ϑ(a(x) jX (x)) = (h ◦ a(x)) ϑ(jX (x)) = (h ◦ a(x)) ϕ(x). (6.4) x∈X
x∈X
x∈X ϕ
(b) (Existence) Referring to (6.4) we introduce a map QX −−−→ Y by ϕ (a) = (h ◦ a(x)) ϕ(x), a ∈ QX .
(6.5)
x∈X
For any x ∈ X we have ϕ (e ·1x ) = h(e) ϕ(x) = ϕ(x). Hence (ϕ , h) makes the diagram (D) commutative. Therefore we have only to show that (ϕ , h) is a fuzzy module homomorphism.
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It follows immediately from the properties (i)–(iii) in Definition 6.1 that (ϕ , h) is a module homomorphism. In order to show that (ϕ , h) is a fuzzy set morphism, we consider the intrinsic R-valued preorder pY in (Y, R, ) and use the property g(y) = pY (y0 , y) for all y ∈ Y . Since for β ∈ R the following chain of equivalences holds: β ≤ g(ϕ (a)) = pY (y0 , ϕ (a)) ⇔ β ϕ (a) ≤ y0 ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ϕ(x) ≤ y0 ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ pY (ϕ(x), y0 ) ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ g(ϕ(x)), I apply the property h ◦ f = g ◦ ϕ and obtain: β ≤ g(ϕ (a)) ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ h ◦ f (x) ⇔β ≤ h ◦ f (x) h ◦ a(x). x∈X
Since h is meet preserving and p is the intrinsic Q-valued preorder in QX , the relation g(ϕ (a)) = h ◦ f (x) h ◦ a(x) x∈X
=h f (x) a(x) x∈X
= h ◦ p(f, a) = h ◦ F (a) follows — i.e. g ◦ ϕ = h ◦ F . Hence (ϕ , h) is a fuzzy set morphism. 2 The assertion of Theorem 6.1 can also be expressed as follows. The forgetful functor from FMod to FQSet has a left adjoint functor. Finally, it can be shown that the category FMod of fuzzy modules is isomorphic to the Eilenberg–Moore category of the fuzzy power set monad. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
G. Birkhoff, Lattice Theory, 3rd edn., Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, Rhode Island, 1995, eight printing. G. Birkhoff, J. Lipson, Heterogeneous algebras, J. Comb. Theory 8 (1970) 115–133. St. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, 1981. R.L. Cignoli, I.M. d’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Springer-Verlag, 1999. P. Eklund, U. Höhle, J. Kortelainen, A survey on the categorical term construction with applications, in: Special Issue Linz 2014, Fuzzy Sets Syst. (2015), submitted for publication. P.I. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963) 115–132. U. Höhle, Editorial, Fuzzy Sets Syst. 40 (1991) 253–256. U. Höhle, Commutative, residuated -monoids, in: U. Höhle, E.P. Klement (Eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, 1995, pp. 53–106. U. Höhle, E.P. Klement, Editorial, Fuzzy Sets Syst. 256 (2014) 1–3. U. Höhle, Modules in the category Sup, dedicated to E.P. Klement, in: R. Mesiar, S. Saminger-Platz (Eds.), Studies in Fuzziness and Soft Computing, Springer-Verlag, 2015/2016. A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Am. Math. Soc. 51 (1984) 309. S. Mac Lane, Categories. For the Working Mathematician, Springer-Verlag, 1971. S. Mace Lane, The health of mathematics, Math. Intell. 5 (4) (1983) 53–56. K. Menger, Ensembles flous et fonctions aléatoires, C. R. Acad. Sci. Paris 232 (1951) 2001–2003. B. Plotkin, Algebra, categories and databases, in: M. Hazewinkel (Ed.), Handbook of Algebra, vol. 2, North-Holland, 2000, pp. 80–148. A. Schmidt, Über deduktive Theorien mit mehreren Sorten von Grunddingen, Math. Ann. 115 (1938) 485–506. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, 1983. L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Fuzzy sets as two-sorted algebras Ulrich Höhle Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Wuppertal, Germany Received 10 October 2014; received in revised form 3 April 2015; accepted 6 May 2015
Abstract This paper makes the attempt to explain the foundations of fuzzy sets from the point of view of universal algebra. The free fuzzy set and the fuzzy power set are constructed. Moreover, fuzzy power sets give rise to a monad. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets. © 2015 Elsevier B.V. All rights reserved. Keywords: Frobenius -monoid; 2-Sorted algebra; Fuzzy set; Fuzzy power set; Fuzzy module
1. Introduction and motivation The fiftieth anniversary of fuzzy sets is a good occasion to go back to their roots and to explain the impact of universal algebra on fuzzy set theory. Therefore I do not consider fuzzy sets as generalized characteristic functions (cf. [7]) or as enriched presheaves (cf. [9]), but simply as two-sorted algebras without any further specific meaning. Before I put down the axioms of this type of algebra it seems to be interesting to consult first the well known paper on fuzzy sets [18] published by L.A. Zadeh in 1965, even though the principal idea of a fuzzy set goes back to K. Menger (cf. [14]). fA In Section 2 of [18] a fuzzy set (class) A in X is not defined, but characterized by a function X −−− → [0, 1] which L.A. Zadeh understands as membership (characteristic) function assigning a real number of the interval [0, 1] to each point in X. Also S. Mac Lane in his critical review on the health of mathematics admits that this idea is attractive and gives the following explanation of this function (cf. p. 54 in [13]): . . . instead of saying that an element x is or is not in the set A, let us measure the likelihood that x is in A.
As I said before I do not continue the debate on membership and measurement of membership, but I understand simply a fuzzy set in X as map from X to [0, 1]. The interesting question is now which algebraic properties of [0, 1] have been used by L.A. Zadeh for his calculus of fuzzy sets. A short glance at [18] shows that he has selected four operations E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2015.05.004 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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– one unary operation sending each element α ∈ [0, 1] to 1 − α, – three binary operations given by the minimum, maximum and product, and a partially defined operation determined by the addition which he calls algebraic sum (cf. p. 344 in [18]). For the contemporary it is unclear why L.A. Zadeh did not complete the algebraic sum to a totally defined binary operation which would coincide with the t -conorm corresponding to Łukasiewicz arithmetic conjunction determined by: α ∗ β = max(α + β − 1, 0),
α, β ∈ [0, 1].
But in a more mathematical language it is clear that L.A. Zadeh has used the standard structure of an MV-algebra on [0, 1] (cf. [4,8]) ([0, 1], min, max, ∗) in which the product · plays the role of a dominating binary operation (cf. [17]) — i.e. (α · β) ∗ (γ · δ) ≤ (α ∗ γ ) · (β ∗ δ),
α, β, γ , δ ∈ [0, 1].
Since MV-algebras are divisible and commutative Frobenius -monoids (cf. [1]), I put down the following abstract definition of a fuzzy set: f
A triple (X, f, Q) is called a fuzzy set iff X is a set, Q is a Frobenius -monoid and X −→ Q is a map.
The aim of this paper is to show that fuzzy sets in the previous sense are two-sorted algebras — i.e. the underlying Frobenius -monoid is incorporated into the definition of a fuzzy set and the membership map f is viewed as a unary two-sorted operation (cf. [6,2]).1 The precise definition of the corresponding two-sorted signature will be given in Section 4. Proceeding in this way the techniques of universal algebra are available, and it is not surprising that free fuzzy sets exist. In particular, every fuzzy set is a quotient of a free fuzzy set. Moreover, in the case of Frobenius quantales the fuzzy power set construction and its monadic basis are explained. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets. As a preparation I first explain some basic properties of Frobenius -monoids. 2. Frobenius -monoids A residuated -semigroup is a quintuple (X, ≤, ∗, , ) where (X, ∗) is a semigroup, (X, ≤) is a lattice with the corresponding binary lattice operations ∧ and ∨, and and are two further binary operations on X satisfying the following condition for all x, y, z ∈ X (cf. [1]): y ≤ x z ⇔ x ∗ y ≤ z ⇔ x ≤ z y.
(AD)
In this context is called the right-implication and is called the left-implication w.r.t. the semigroup operation ∗. The next proposition collects some fundamental properties of residuated -semigroups. Proposition 2.1. Let ((X, ≤), ∗, , ) be residuated -semigroup. Then the semigroup operation ∗ is distributive on binary joins in each variable separately — i.e. x ∗ (y ∨ z) = (x ∗ y) ∨ (x ∗ z)
and (x ∨ y) ∗ z = (x ∗ z) ∨ (y ∗ z).
(D)
If ⊥ is the universal lower bound of (X, ≤), then ⊥ is always the zero element of (X, ∗). Moreover the following properties hold for all x, y, z ∈ X: (i) (x ∨ y) z = (x z) ∧ (y z) and z (x ∨ y) = (z x) ∧ (z y). (ii) x (y ∧ z) = (x y) ∧ (x z) and (z ∧ y) x = (z x) ∧ (y x). 1 It is interesting to see that the term many-sorted has its origin in mathematical logic (cf. [16]) and plays now a prominent role in certain areas of algebra (cf. [15]).
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(iii) (x ∗ y) z = y (x z) and z (y ∗ x) = (z x) y. (iv) (x y) z = x (y z). Even though the previous proposition is well known (see e.g. Theorem 4 on p. 326 in [1]), I give here a direct proof of Proposition 2.1 for the convenience of the reader. Proof. Because of the reflexivity and transitivity axiom of partial orders the isotonicity of the semigroup operation ∗ is an immediate corollary of condition (AD). Hence with regard to (D) I have only the verify the inequalities x ∗ (y ∨ z) ≤ (x ∗ y) ∨ (x ∗ z)
and
(x ∨ y) ∗ z ≤ (x ∗ z) ∨ (y ∗ z).
For this purpose I put u = (x ∗ y) ∨ (x ∗ z) and infer from (AD) that y ∨ z ≤ x u holds. Now I apply again (AD) and obtain x ∗ (y ∨ z) ≤ u. Replacing the right-implication by the left-implication the verification of (x ∨ y) ∗ z ≤ (x ∗ z) ∨ (y ∗ z) follows by means of an analogous argument. Finally, I assume that the universal lower bound ⊥ exists in (X, ≤), and fix an arbitrary element x ∈ X. Since ⊥ ≤ x z holds for all z ∈ X, I conclude form (AD) that x ∗ ⊥ coincides with the universal lower bound. Analogously I verify ⊥ ∗ x = ⊥. Hence ⊥ is the zero element of (X, ∗). Now let us turn to the properties (i)–(iv). Because of the isotonicity of ∗ it is easily seen that the inequalities (x ∨ y) z ≤ (x z) ∧ (y z) and z (x ∨ y) ≤ (z x) ∧ (z y) hold. Since ∗ is distributive on binary joins, one infers from (AD) that also the converse inequalities are valid. Hence (i) is verified. The relation (ii) is an immediate corollary from the isotonicity of ∗. Finally, (iii) and (iv) follow from the associativity of ∗. In fact, the following chain of equivalences holds true: v ≤ (x ∗ y) z ⇔ x ∗ y ∗ v ≤ z ⇔ y ∗ v ≤ x z ⇔ v ≤ y (x z). Analogously the relation z (y ∗ x) = (z x) y can be verified. The subsequent chain of equivalences v ≤ (x y) z ⇔ v ∗ z ≤ x y ⇔ x ∗ v ∗ y ≤ z ⇔ x ∗ v ≤ z y ⇔ v ≤ x (z y) ensures the validity of (v).
2
An element d of a residuated -semigroup (X, ≤, ∗, , ) is called dualizing, if for all x ∈ X the following relations hold: d (x d) = x
and
(d x) d = x.
(2.1)
Proposition 2.2. Let (X, ≤, ∗, , ) be a residuated -semigroup with a dualizing element d. Then the following properties hold: (i) d d is a right-unit and d d is a left-unit w.r.t. ∗ — this means x ∗ (d d) = x and (d d) ∗ x = x for all x ∈ X. (ii) d d and d d coincide and form the unit of (Q, ∗). Moreover, if (X, ≤) be a bounded lattice with universal bounds ⊥ and , then the additional properties are valid (iii) d = ⊥ = d and ⊥ d = = d ⊥. Proof. In order to show that d d is a right-unit of (X, ∗) I fix x ∈ X and use the property (iii) in Proposition 2.1 and the dualizing property of d:
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x ∗ (d d) = d (x ∗ (d d)) d = d (d d) x d = (d x) d = x. Hence d d is a right-unit in (X, ∗). Analogously we verify that (d d) is a left-unit in (Q, ∗). Finally, (ii) follows immediately from (i). Now we assume that (X, ≤) has universal bounds. Since ⊥ is the zero element of (X, ∗) (cf. Proposition 2.1) the relations ⊥ d = and d ⊥ = hold. Hence (iii) follows from (2.1). 2 The previous proposition shows that every residuated -semigroup with a dualizing element is necessarily unital. This observation is a motivation to introduce the following terminology. Definition 2.1. A sextuple (X, ≤, ∗, , , d) is called a Frobenius -monoid, if (X, ≤, ∗, , ) is a residuated -semigroup and d is a dualizing element in (X, ≤, ∗, , ). The next proposition explains first properties of Frobenius -monoids. Proposition 2.3. In every Frobenius -monoid (X, ≤, ∗, , , d) the following properties hold for all x, z ∈ X: (i) (ii) (iii) (iv)
x z = ((d z) ∗ x) d. z x = d (x ∗ (z d)). (x ∧ z) d = (x d) ∨ (z d). d (x ∧ z) = (d x) ∨ (d z).
Proof. Because of (2.1) and the property (iii) in Proposition 2.1 the relation x z = x ((d z) d) = ((d z) ∗ x) d holds Hence (i) is verified. Further, (iii) follows from the following relation: (x ∧ z) d = d (x d) ∨ (z d) d = (x d) ∨ (z d). Finally, (ii) and (iv) can be verified analogously. 2 Left-sided and right-sided elements play a special role in Frobenius -monoids. For this purpose I recall the following terminology (cf. [1]). Definition 2.2. Let (X, ≤, ∗, , ) be a residuated -semigroup such that the universal upper bound exists in (X, ≤). An element x ∈ X is called left-sided (right-sided) if ∗ x ≤ x (x ∗ ≤ x) holds. An element x of X is said to be two-sided if x is left-sided and right-sided. Lemma 2.4. Let (X, ≤, ∗, , , d) be a Frobenius -monoid such that (X, ≤) is a bounded lattice. Then the following properties hold: (i) If x ∈ X is left-sided, then x d = x ⊥ holds. (ii) If x ∈ X is right-sided, then d x = ⊥ x holds. Proof. In order to verify (i) we proceed as follows. Let z be a left-sided element of X with z ≤ d. Then d z = holds. Hence z = ⊥ follows from Proposition 2.2(iii). Now we choose an arbitrary left-sided element x. Since ∗ is associative, x ∗ (x d) is left-sided, and we deduce from x ∗ (x d) ≤ d that x ∗ (x d) = ⊥ is valid. Hence x d = x ⊥ follows. — Analogously we prove (ii). 2
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Corollary 2.5. Every Frobenius -monoid with universal bounds satisfies the Frobenius property — i.e. for all leftsided elements x and for all right-sided elements y the following relation holds: x = ⊥ (x ⊥) and
y = (⊥ y) ⊥.
(Frobenius Property)
Proof. Let (X, ≤, ∗, , , d) be a Frobenius -monoid with universal bounds and let x ∈ X be right-sided. Then the relation d x = ⊥ x follows from Lemma 2.4(ii). Since ⊥ is the zero element of (X, ∗) (cf. Proposition 2.1), the element ⊥ x is left-sided. Now I use Lemma 2.4(i) and obtain: x = (d x) d = (⊥ x) d = (⊥ x) ⊥. If y is left-sided, then the property y = ⊥ (y ⊥) can be verified analogously. 2 Let (X, ≤, ∗, , , d) and (X, ≤, ∗, , , d) be Frobenius -monoids. A map X −−h−→ Y is called a Frobenius morphism, if h is a residuated -semigroup homomorphism and preserves the respective dualizing elements — i.e. h(x ∧ y) = h(x) ∧ h(y),
h(x y) = h(x) h(y),
h(x ∗ y) = h(x) ∗ h(y),
h(x ∨ y) = h(x) ∨ h(y),
h(y x) = h(y) h(x),
h(d) = d.
Since the unit in any Frobenius -monoid with dualizing element d has the form d d (cf. Proposition 2.2(ii)), it follows that Frobenius morphisms are always unital. Because of Proposition 2.3 Frobenius morphisms can be characterized as follows. Proposition 2.6. Let (X, ≤, ∗, , , d) and (Y, ≤, ∗, , , d) be Frobenius -monoids, and let map X −−h−→ Y be a map. Then the following assertions are equivalent: (i) h is a Frobenius morphism. (ii) h preserves the respective dualizing elements and satisfies the properties h(x ∗ z) = h(x) ∗ h(z), h(x d) = h(x) d, h(d z) = d h(z), h(x ∨ z) = h(x) ∨ h(z) for all x, z ∈ X. Proof. The implication (i) ⇒ (ii) is evident. On the other hand, if (ii) holds, then we infer form Proposition 2.3(i) and Proposition 2.3(ii) that h preserves the left- and right-implication. Finally, we use Proposition 2.3(iv) and Proposition 2.1(i) and obtain: h(x ∧ z) = h(d (x ∧ z)) d = h((d x) ∨ (d z)) d = h(d x) ∨ h(d z) d = (d h(x)) d ∧ (d h(z)) d = h(x) ∧ h(z). Hence h also preserves the binary meet operation. 2 The class of Frobenius -monoids and Frobenius morphisms form a category Frob in an obvious way. 3. Frobenius -monoids as algebras It is well known (cf. [3]) that every lattice (X, ≤) forms an algebra (X, , ⊕) where and ⊕ are commutative and associative, binary operations on X satisfying the absorption laws: x (x ⊕ y) = x
and
x ⊕ (x y) = x
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for all x, y ∈ X. In this context plays the role of binary meets and ⊕ the role of binary joins. Further, any semigroup is an algebra with a unique associative binary operation. Hence in order to understand Frobenius -monoids as algebras it is only necessary to formulate an equational characterization of condition (AD). Theorem 3.1. Let (X, ≤) be a lattice provided with an associative and binary operations ∗. Further, let and two further binary operations on X. Then the following assertions are equivalent: (i) The quintuple (X, ≤, ∗, , ) is a residuated -semigroup. (ii) The following properties hold: (a) z = z ∨ (x ∗ ((x z) ∧ y)), y = y ∧ (x ((x ∗ y) ∨ z)), x, y, z ∈ X. (b) z = z ∨ (((z x) ∧ y) ∗ x), y = y ∧ (((y ∗ x) ∨ z) x), x, y, z ∈ X. Proof. If (i) holds, then the properties (a) and (b) follow from the isotonicity of the semigroup operation ∗ (cf. (D)), the property (ii) in Proposition 2.1 and the following observations: x ∗ (x z) ≤ z,
y ≤ x (x ∗ y),
(z x) ∗ x ≤ z,
y ≤ (y ∗ x) x).
On the other hand, the property (a) implies the equivalence x ∗ y ≤ z ⇔ y ≤ x z. In fact, the relation x ∗ y ≤ z implies y = y ∧ (x z) — i.e. y ≤ x z, and the relation y ≤ x z implies z = z ∨ (x ∗ y) — i.e. x ∗ y ≤ z. Analogously, it can be shown that the property (b) implies the equivalence y ∗ x ≤ z ⇔ y ≤ z x. Hence (ii) implies the condition (AD). 2 Referring to (2.1) a Frobenius -monoid is an algebra of the following form (X, , ⊕, ∗, , , d) where d is a nullary operation (i.e. a constant) of X and , ⊕, ∗, , are binary operations on X satisfying the following axioms: (1) (2) (3) (4) (5) (6)
, ⊕ and ∗ are associative operations. and ⊕ are commutative operations. x (x ⊕ y) = x and x ⊕ (x y) = x. z = z ⊕ (x ∗ ((x z) y)), y = y (x ((x ∗ y) ⊕ z)). z = z ⊕ (((z x) y) ∗ x), y = y (((y ∗ x) ⊕ z) x). x = d (x d), and x = (d x) d.
(Absorption Laws)
Hence the (one-sorted) signature = (n )n∈N0 of a Frobenius -monoid has the form: 0 = {ω0 }, 1 = ∅, 2 = {ω1 , ω2 , ω3 , ω4 , ω5 }, n = ∅ for 3 ≤ n. Then we conclude from the (one-sorted) term construction in Set (cf. [3,5]) that for every set X the free -algebra X exists and is given by the set of all -terms in X. In order to arrive at the free Frobenius -monoid generated by X it is necessary to form an appropriate quotient of the free -algebra X fulfilling the equation in (1)–(6). Hence the subsequent theorem holds. Theorem 3.2. Let X be a set. Then there exists a Frobenius -monoid (X , ≤ , ∗ , , , d )
and a map
η
X X −−− → X
such that for any further Frobenius -monoid (Y, ≤, ∗, , , d) and for any further map X −−h−→ X there exists a
unique Frobenius morphism X −−h−→ Y making the following diagram commutative: X
ηX
X
h
h
Y
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Theorem 3.2 can also expressed as the statement that the forgetful functor from Frob to Set has a left adjoint. In this sense the choice of morphisms between Frobenius -monoids was not free in Section 2, but dictated by methods of universal algebra. 4. Fuzzy sets as two-sorted algebras Now I take up the abstract definition of a fuzzy set given in Section 1 and recall that a fuzzy set is a triple (X, f, Q) f where X is a set, Q is a Frobenius -monoid, and X −→ Q is a map. Since Frobenius monoids are algebras (cf. Section 3), it is evident that fuzzy sets are two-sorted algebras. Hence a morphism from a fuzzy set (X, f, Q) to a ϕ h R is a Frobenius morphism fuzzy set (Y, g, R) is a pair of maps (ϕ, h) where X −→ Y is a simple map and Q −→ such that the following diagram is commutative in Set: ϕ
X
Y g
f
Q
h
R
The category of fuzzy sets and their morphisms is denoted by FSet. In the following consideration I specify the two-sorted signature of a fuzzy set. For this purpose I recall the free monoid M generated by {1, 2} — i.e. M = {(0, ∅)} ∪
{n} × {1, 2}n .
n∈N (σ,s)
Then a two-signature = (, {1, 2}) is a system (n )(n,σ,s)∈M×{1,2} of sets indexed by M × {1, 2}. Elements (σ,s) are also called two-sorted functional symbols of arity n. Because of this semantic meaning the following of n (σ,s) notation for n is used in computer science: (∅,s)
0
= →s 0 ,
(σ,s) = sn1 ×...×sn →s , n
1 ≤ n, σ = (s1 , . . . , sn ).
After these preparations I specify the two-sorted signature F = (, {1, 2}) of a fuzzy set: →1 0 = ∅, →2 0 = {ω0 }, 1→2 = {ω1 2 }, 1 2×2→2 = {ω1 , ω2 , ω3 , ω4 , ω5 }, 2 sn1 ×...×sn →s = ∅,
elsewhere.
Then the two-sorted term construction in Set shows that for any pair (X1 , X2 ) of sets X1 and X2 the free -algebra of F -terms in (X1 , X2 ) exists (see e.g. [5]). Subsequently, an appropriate quotient of the free F -algebra leads to the free fuzzy set generated by (X1 , X2 ). In the following consideration I give a direct proof of this standard fact. Theorem 4.1. Let (X1 , X2 ) be a pair of sets and X1 X2 be the disjoint join of X1 and X2 with the canonical jX
jX
1 2 embeddings X1 −−→ X1 X2 and X2 −−→ X1 X2 . Further, let Q be the free Frobenius -monoid generated by ηX1 X2 X1 X2 with the embedding X1 X2 −−−−−→ Q. Then (X1 , π, Q) with π = ηX1 X2 ◦ jX1 is a fuzzy set. Moreover, for ϕ h any further fuzzy set (Y, g, R) and for any further pair (ϕ, h) of maps X1 −→ Y and X2 −→ R there exists a unique (ψ,k) fuzzy set morphism (X, π, Q) −−−−→ (Y, g, R) making the following diagram commutative in Set × Set:
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8
(X1 , X2 )
(1X1 ,ηX X2 ◦jX2 )
(X1 , Q)
(ψ,k)
(ϕ,h)
(E)
(Y, R) — i.e. ψ = ϕ and h = k ◦ (ηX X2 ◦ jX2 ). Proof. (a) (Uniqueness) Let us assume that (ψ, k) is a fuzzy set morphism making the diagram (E) commutative. Hence ψ =ϕ
and k ◦ (ηX X2 ◦ jX2 ) = h
(4.1)
follow. Since (ψ, k) is a fuzzy set morphism from (X1 , π, Q) to (Y, g, R), the relation k ◦ ηX1 X2 ◦ jX1 = k ◦ π = g ◦ ϕ
(4.2)
holds. Let us recall that the coproduct in Set is given by disjoint unions. Hence there exists a unique map X 1 X2
(g◦ϕ)h
R satisfying the properties (g ◦ ϕ) h) ◦ jX1 = g ◦ ϕ and (g ◦ ϕ) h) ◦ jX2 = h. In particular, the relation k ◦ ηX1 X2 = (g ◦ ϕ) h follows from (4.1) and (4.2). Since k is a Frobenius morphism, I conclude from Theorem 3.2 that k coincides with the unique extension of (g ◦ ϕ) h to a Frobenius morphism from Q to R. Hence the uniqueness of (ψ, k) is verified. (b) (Existence) Let (g ◦ ϕ) h be the unique extension of (g ◦ ϕ) h to a Frobenius morphism from Q to R (cf. Theorem 3.2). Then the argumentation in (a) suggests to define (ψ, k) as follows: ψ = ϕ and k = (g ◦ ϕ) h . We have to show that (ψ, k) is a fuzzy set morphism making the diagram (E) commutative. First it is easily seen that the following relation holds: k ◦ π = (g ◦ ϕ) h ◦ ηX1 X2 ◦ jX1 = (g ◦ ϕ) h ◦ jX1 = g ◦ ϕ = g ◦ ψ. Hence (ψ, k) is a fuzzy set morphism. A further observation is the following one: k ◦ (ηX1 X2 ◦ jX2 ) = (g ◦ ϕ) h ◦ ηX1 X2 ◦ jX2 = (g ◦ ϕ) h ◦ jX2 = h. Hence the diagram (E) is commutative. 2 The previous theorem can also be expressed by the statement that the forgetful functor from the category FSet of fuzzy sets to Set × Set has a left adjoint. In particular, in the sense of FSet every fuzzy set is a quotient of an appropriate free fuzzy set. 5. Fuzzy power set monad In this section I assume that the underlying lattice of some given Frobenius -monoid is complete — this means that Q is a Frobenius quantale. In this context I recall that the left-implication and right-implication of a Frobenius quantale have the following form:
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β α =
{γ ∈ Q | γ ∗ α ≤ β} and α β =
9
{γ ∈ Q | α ∗ γ ≤ β}.
Further, every Frobenius quantale Q with the dualizing element d satisfies the following properties A d = {a d | a ∈ A} and d A = {d a | a ∈ A}
(5.1)
for all subsets A of Q. In fact, if A is empty, then (5.1) follows from Proposition 2.2(iii). In the case of non-empty subsets the proof of Proposition 2.3(iii) and (iv) carries over. Morphisms between Frobenius quantales are Frobenius morphisms which are join preserving. Referring to (5.1) it is easily seen that join preserving Frobenius morphisms are always meet preserving. Moreover, the MacNeille completion of every Frobenius -monoid is a Frobenius quantale (for the commutative case see [8]), and every Frobenius morphism having a right adjoint has a unique extension to a join preserving Frobenius morphism between the corresponding Frobenius quantales. Let FQSet be the subcategory of FSet consisting of fuzzy sets (X, f, Q) and fuzzy set morphisms (ϕ, h) s.t. Q is a Frobenius quantale and h is a join preserving. The aim of this section is to construct a monad on FQSet which can be understood as the fuzzy power set monad. First we define an object function P: |FQSet| → |FQSet| as follows. With every fuzzy set (X, f, Q) we associate a fuzzy set P(X, f, Q) := (QX , F, Q) where the map QX −F→ Q is determined by: F (g) = f (x) g(x), g ∈ QX . (5.2) x∈X
The fuzzy set P(X, f, Q) is also called the fuzzy power set of (X, f, Q). Now I complete the object function P (ϕ,h) to an endofunctor of FQSet as follows. For every fuzzy set morphism (X, f, Q) −−−→ (Y, g, R) I define a pair P(ϕ, h) := (P (ϕ), h) of maps by P (ϕ)(m)(y) = {h ◦ m(x) | ϕ(x) = y}, m ∈ QX , y ∈ Y (5.3) and show that (P (ϕ), h) is a fuzzy set morphism from P(X, f, Q) to P(Y, g, R). For this purpose I fix m ∈ QX . Since h is meet preserving, I obtain: h ◦ F (m) = h(f (x)) h(m(x)) x∈X
=
g(ϕ(x)) h(m(x))
x∈X
=
g(y)
y∈Y
=
{h(m(x))|ϕ(x) = y}
g(y) P (ϕ)(m)(y)
y∈Y
= G(P (ϕ)(m)). Hence P(ϕ, h) is a fuzzy set morphism. It is easily seen that P preserves the composition in FQSet. Finally we introduce two natural transformations η: idFQSet → P and μ: P ◦ P → P. For this purpose the unit (resp. universal lower bound) of the respective Frobenius quantales is denoted by e (resp. ⊥): η(X,f,Q) =(jX ,1Q )
(X, f, Q) jX (x) = e · 1x Because of f (x) =
P(X, f, Q) e, z = x and e · 1x (z) = ⊥, z = x,
z∈X f (z) e · 1x (z) = F (e · 1x )
where x, z ∈ X. it is evident that (jX , 1Q ) is a fuzzy set morphism. Further the (ϕ,h)
relation P (ϕ)(e · 1x ) = h(e) · 1ϕ(x) holds for every fuzzy set morphism (X, f, Q) −−−→ (Y, g, R) where I recall that h(e) is the unit in R. Hence η with components η(X,f,Q) is a natural transformation from idFQSet to P.
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The second natural transformation μ with components μ(X,f,Q) =(νX ,1Q )
P(P(X, f, Q)) has the form: νX (K)(x) =
P(X, f, Q)
X
K(k) ∗ k(x),
K ∈ Q(Q ) .
k∈QX
Because of f (x) νX (K)(x) = f (x) K(k) ∗ k(x) x∈X
x∈X
k∈QX
=
f (x) k(x) K(k)
x∈X
k∈QX
X
the pair (νX , 1Q ) is a fuzzy set morphism. Moreover for all K ∈ Q(Q ) and y ∈ Y the following relation holds: [νY (P (P (ϕ))(K))](y) = {h ◦ K(k) | P (ϕ)(k) = ∗ (y) ∈QY
=
h ◦ K(k) ∗
{h ◦ k(x) | ϕ(x) = y}
k∈QX
=
{h K(k) ∗ k(x)) | ϕ(x) = y}
k∈QX
= [P (ϕ)(νX (K))](y). Hence μ with components μ(X,f,Q) is a natural transformation from P ◦ P to P. Theorem 5.1. The triple (P, η, μ) is a monad on FQSet. Proof. Let (X, f, Q) be a fuzzy set. Referring to [12] I have to show the commutativity of the following diagrams: P(X, f, Q)
ηP(X,f,Q)
P(P(X, f, Q))
P (η(X,f,Q) )
μ(X,f,Q)
1P(X,f,Q)
P(X, f, Q)
1P(X,f,Q)
(Unit)
P(X, f, Q) P(P(P(X, f, Q)))
P (μ(X,f,Q) )
μP(X,f,Q)
P(P(X, f, Q))
P(P(X, f, Q))
μ(X,f,Q)
μ(X,f,Q)
(Associativity)
P(X, f, Q)
Let k be an element of QX . It is easily seen that the subsequent relations hold: νX (e · 1k ) = k, [P (jX )(k)]() = {k(x) | e · 1x = }, νX (P (jX )(k)) = k. Hence the unit axiom is verified. Since for K ∈ Q(Q
QX )
and x ∈ X one obtains:
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U. Höhle / Fuzzy Sets and Systems ••• (••••) •••–•••
K(K) ∗ K(k) ∗ k(x) =
k∈QX , K∈Q
QX
K∈Q
11
K(K) ∗ νX (K)(x)
QX
= νX (P (νX )(K))(x), the associativity axiom holds also true. 2 The previous theorem shows that a non-trivial “power set theory” is available for fuzzy sets viewed as two-sorted algebras. This observation is a motivation to call the monad in Theorem 5.1 the fuzzy power set monad. 6. Fuzzy modules In this section I show that the fuzzy power set is the free fuzzy module generated by its underlying fuzzy set. First I need some terminology. Definition 6.1. A left module on a Frobenius quantale is a triple (X, Q, ) where X is a complete lattice, Q is a Frobenius quantale and is a left action on X w.r.t. Q in the sense of the category Sup of all complete lattices and ⊗ join preserving maps — this means that Q × X −→ X is an outer binary operation on X satisfying the following axioms (cf. [11,10]):
(i) The map Q × X −−−→ X is join preserving in each variable separately — i.e. αi x = αi x and α ( xi = α xi . i∈I
i∈I
i∈I
i∈I
(ii) α (β x) = (α ∗ β) x where ∗ denotes the multiplication in Q. (iii) e x = x where e is the unit in Q. Let (X, Q, ) and (Y, R, ) be left modules on Frobenius quantales. A module homomorphism from (X, Q, ) to ϕ h R s.t. ϕ is join preserving, h is a join preserving Frobenius morphism, (Y, R, ) is a pair of maps X −→ Y and Q −→ and the following diagram is commutative: Q×X
h×ϕ
R×Y
X
ϕ
Y
Left modules on Frobenius quantales and morphism between them form a category in an obvious way. From the point of view of many valued mathematics the interesting point is here that any left module (X, Q, ) has an intrinsic Q-valued preorder pX determined by: pX (x2 , x1 ) = {α ∈ Q | α x1 ≤ x2 }, x1 , x2 ∈ X. (6.1) This observation motives the following definition. Definition 6.2. A quintuple (X, x0 , f, Q, ) is a fuzzy left module if x0 is an element of X, (X, f, Q) is a fuzzy set and (X, Q, ) is a left module such that the property f (x) = {α ∈ Q | α x ≤ x0 } holds for all x ∈ X — i.e. f (x) coincides with the value pX (x0 , x) of the intrinsic Q-valued preorder at (x0 , x) for all x ∈ X (cf. (6.1)). A pair (ϕ, h) is a fuzzy module homomorphism iff (ϕ, h) is a fuzzy set morphism and a module homomorphism. It is easily seen that fuzzy modules and fuzzy module homomorphisms form a category FMod. The next example explains that every fuzzy power set gives rise to a fuzzy module.
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12
Example 6.1. Let (X, f, Q) be a fuzzy set and P(X, f, Q) = (QX , F, Q) be its fuzzy power set. Further, let ∗ be the multiplication in Q and be the left action on QX defined by (α g)(x) = α ∗ g(x),
α ∈ Q,
x ∈ X,
g ∈ QX .
(6.2)
Then (QX , Q, ) is a left module. Obviously, the intrinsic Q-valued preorder p on QX has the form (cf. (6.1)): p(k, g) = {α ∈ Q | α g ≤ k} = k(x) g(x), g, k ∈ QX . x∈X
Hence we conclude from the definition of F (cf. (5.2)) that the fuzzy power set P(X, f, Q) can be enriched to a fuzzy left module as follows: (QX , f, F, Q, ). In this sense every fuzzy power set can be understood as a fuzzy left module where is given by (6.2). Now we prove the main theorem of this section which says that the fuzzy power set is the free fuzzy module generated by its underlying fuzzy set. (ϕ,h)
Theorem 6.1. Let (X, f, Q) be a fuzzy set and (Y, y0 , g, R, ) be a fuzzy left module. Further, let (X, f, Q) −−−→ (ϕ ,h )
(Y, g, R) be a fuzzy morphism. Then there exists a unique fuzzy module homomorphism (QX , f, F, Q, ) −−−−−→ (Y, y0 , g, R, ) such that the following diagram is commutative in FQSet: (X, f, Q)
η(X,f,Q)
P(X, f, Q) (D)
(ϕ ,h )
(ϕ,h)
(Y, g, R) In particular, h and h coincide. Proof. Let us begin with some general observations. Since the second component of η(X,f,Q) is the identity of Q, the join preserving Frobenius morphisms h and h coincide. Moreover, every element a ∈ QX can be represented as follows: a= a(x) jX (x) (6.3) x∈X
where is determined by (6.2) and jX has the form (cf. Section 5): jX (x) = e · 1x ,
e is the unit of Q.
(a) (Uniqueness) Let (ϑ, k) be a fuzzy module homomorphism making the diagram (D) commutative. Hence h and k coincide. In order to show that ϑ is uniquely determined, we use the fact that (ϑ, h) is a fortiori a module homomorphism and derive the following relation form (6.3): ϑ(a) = ϑ(a(x) jX (x)) = (h ◦ a(x)) ϑ(jX (x)) = (h ◦ a(x)) ϕ(x). (6.4) x∈X
x∈X
x∈X ϕ
(b) (Existence) Referring to (6.4) we introduce a map QX −−−→ Y by ϕ (a) = (h ◦ a(x)) ϕ(x), a ∈ QX .
(6.5)
x∈X
For any x ∈ X we have ϕ (e ·1x ) = h(e) ϕ(x) = ϕ(x). Hence (ϕ , h) makes the diagram (D) commutative. Therefore we have only to show that (ϕ , h) is a fuzzy module homomorphism.
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It follows immediately from the properties (i)–(iii) in Definition 6.1 that (ϕ , h) is a module homomorphism. In order to show that (ϕ , h) is a fuzzy set morphism, we consider the intrinsic R-valued preorder pY in (Y, R, ) and use the property g(y) = pY (y0 , y) for all y ∈ Y . Since for β ∈ R the following chain of equivalences holds: β ≤ g(ϕ (a)) = pY (y0 , ϕ (a)) ⇔ β ϕ (a) ≤ y0 ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ϕ(x) ≤ y0 ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ pY (ϕ(x), y0 ) ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ g(ϕ(x)), I apply the property h ◦ f = g ◦ ϕ and obtain: β ≤ g(ϕ (a)) ⇔ ∀ x ∈ X : β ∗ (h ◦ a(x)) ≤ h ◦ f (x) ⇔β ≤ h ◦ f (x) h ◦ a(x). x∈X
Since h is meet preserving and p is the intrinsic Q-valued preorder in QX , the relation g(ϕ (a)) = h ◦ f (x) h ◦ a(x) x∈X
=h f (x) a(x) x∈X
= h ◦ p(f, a) = h ◦ F (a) follows — i.e. g ◦ ϕ = h ◦ F . Hence (ϕ , h) is a fuzzy set morphism. 2 The assertion of Theorem 6.1 can also be expressed as follows. The forgetful functor from FMod to FQSet has a left adjoint functor. Finally, it can be shown that the category FMod of fuzzy modules is isomorphic to the Eilenberg–Moore category of the fuzzy power set monad. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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