1−
where once againεis a positive infinitesimal, and the monoidal operation?is defined by (1−
For these monoids and for others we will exhibit a more effective embedding into [0,1]in the next sections.
5.4 Embedding finite lexicographical products
First we present an example with detailed explanations. Careful reading of it will help the under-standing the main theorem of this section.
Example 5.7 (Definition1 of the t-norm(TP)h+i.) Letε >0be an infinitesimal, X ={1−n·ε−r·ε2 |n ∈N, r ∈R+}.
This set is dual-isomorphic in the obvious way to the setN×R+equipped with the lexicographic order. First, we are going to construct an order-preserving isomorphismη fromX to]0,1]such thatη(1−ε) =α1,η(1−ε2) =α2 (α1andα2are fixed elements of]0,1[). Then we should have α1 < α2 by the order-preserving nature of η. Such an isomorphism should, of course, satisfy limn→∞η(1−n ·ε) = 0 and limn→∞η(1−n ·ε2) = η(1−ε). We shall define η such that η(1−n·ε) = αn1 holds; this obeys the above mentioned first requirement. In order to simplify the forthcoming formulas we introduce a1 = α1, a2 = ϕα1(α2), and φn = ϕ−1a1,n. Now define
1The t-norm of this example has been discovered independently by H´ajek in [54].
Next, we define a binary operation⊕TP onR+(the dual ofTP) byr⊕TPs = loga2(TP(ar2, as2)). com-mutative, residuated integral`-monoid. Therefore, its isomorphic copy on]0,1](viaη) is a left-continuous t-norm without zero divisors:
Let us define a binary operation on]0,1]by
(TP)h+i(x, y) =η η−1(x)∗◦η−1(y) .
(TP)h+ican equivalently be described by the following two other formulations:
(TP)h+i(x, y) = φn(x)+n(y) a2r(x)⊕TPr(y)
Note that the value of(TP)h+idoesn’t depend on the particular choice ofα2, as it is seen from the last formulation.
The first definition of(TP)h+iemphasizes that(TP)h+iisisomorphicto a commutative, resid-uated, integral `-monoid on a lexicographic product space. The second one is an analytical description, the functions n andr can be computed according to the given formulas. The third one is a definition with arecursiveflavor, as we will see in Theorem 5.8.
0 0.2 0.4 0.6 0.8 1
We shall generalize the above example by replacing the lexicographic productN×R+byN× . . .×N×R+, the addition of natural numbers + by⊕i’s which are commutative, disjunctive
`-monoids on N with zero 0, and the product t-norm TP by any t-norm without zero divisors.
Instead of the notation Th⊕k,...,⊕1i we will use in the theorem the shorter Thki notation for the resulted t-norm.
5.4. EMBEDDING FINITE LEXICOGRAPHICAL PRODUCTS 91 Theorem 5.8 (Thki) Let k ∈ N and T be any t-norm without zero divisors. Let ⊕i be a commutative, disjunctive`-monoidal operation onNwith zero0for1≤i≤k.
i. The three definitions forThkigiven in 1, 2 and 3 below are equivalent.
1. Letε >0be an infinitesimal, Xk = Thenηkis an order-preserving bijection fromXkto]0,1]. Finally, define a binary operation Thkion]0,1]by
In addition,
ii. Thkiis a t-norm without zero divisors.
iii. Thkiis left-continuous if and only if so doesT. iv. Thkiis strictly increasing if and only if so doesT.
v. By using the definition in 3. we have thatThki
]α1,1] is order-isomorphic toThk−1i.
Proof. Claim: ηkis an order-preserving bijection fromXkto]0,1], and its inverse is given by ηk−1(x) = 1−
k+1
X
i=1
nk,i(x)·εi. (5.5)
The proof of the claim is given in the following three items.
a. Lett= 1−Pk+1 which justifies the statement fork= 0. Fork >0we have
ηk(t) =ηk 1−
5.4. EMBEDDING FINITE LEXICOGRAPHICAL PRODUCTS 93 All the ai’s are in ]0,1]. Thus,ηk is clearly strictly increasing. The proof of the claim is concluded.
(5.5) immediately verifies the equivalence between (5.2) and (5.3).
We shall prove the equivalence between (5.3) and (5.4). Letk = 0. We obtain from (5.3) Th0i(x, y) = a1n0,1(x)⊕Tn0,1(y)=a
which confirms the statement fork = 0. Letk = 1. From (5.4) by using the definition ofx1,1, the definition ofn1,2, and the definition of⊕T, we obtain
Th1i(x, y) = ϕ−1a
which is just Th1i(x, y) defined by (5.3). This verifies the stated equivalence for k = 1. Sup-pose that the stated equivalence holds for k−1. That is, α1, . . . , αk, ⊕1, . . . ,⊕k are fixed, and Thk−1i(x, y)defined by (5.4) equals to
φnk−1,1(x)⊕1nk−1,1(y),nk−1,2(x)⊕2nk−1,2(y),...,nk−1,k−1(x)⊕k−1nk−1,k−1(y) aknk−1,k(x)⊕Tnk−1,k(y)
(5.7) Let α˜i = αi−1, ⊕˜i = ⊕i−1 for 2 ≤ i ≤ k + 1. Fix α˜1 arbitrarily in ]0,α˜2[, and let ⊕˜1 be commutative, disjunctive `-monoid on N. Define Thki by (5.3) and (5.4) but based on k, 0 =
˜
α0 < α˜1 < α˜2 < . . . < α˜k < α˜k+1 < 1 and ⊕˜i (1 ≤ i ≤ k). That is, let ˜ai = ϕα˜i−1( ˜αi) (1≤ i≤ k+ 1), and use˜everywhere in the rest of the notations of the definitions. Due to the shift of the indices we have⊕T˜ =⊕T, and
nk−1,i(˜xk,1) = ˜nk,i+1(x) (5.8)
as it is easy to verify. Therefore, by (5.4), using the definition of x˜k,1, the hypothesis of the induction in (5.7) and, finally, (5.8) we obtain thatThki(x, y) =
= ϕ−1
which is justThki(x, y) by (5.3). Thus, the equivalence between (5.3) and (5.4) is verified, and the proof ofiis concluded.
It is a matter of straightforward verification that(Xk,∗◦)is a commutative, integral`-monoid.
In addition, (Xk,∗◦) is residuated if and only if T is residuated, cancellative if and only if T is cancellative. Sinceηkis an isomorphism these proveii–iv. Finally, observe thatα1 =a1, and that nk,1(x) = 0whenx ∈]α1,1]. Therefore, (5.4) shows thatϕα1 is an order-isomorphism between Thki
]α1,1] andThk−1i.
Remark 5.9 IfT is only a left-continuous t-subnorm in Theorem 5.8 (and not a t-norm), or if0is not the zero of⊕i’s but we suppose0⊕i0 = 0then everything holds true in Theorem 5.8 except the boundary condition of the resulted t-norm. Then we obtain t-subnorms.
Remark 5.10 As far as we can see Theorem 5.11 can not be extended so thatT is a t-norm withzero divisors without loosing either the associativity or the left-continuity (that is the resid-uated nature) of the resulted structure.
Corollary 5.11 (Th⊕i) Let T be any t-norm without zero divisors, ⊕ be any commutative, disjunctive`-monoid on Nwith zero 0, a ∈]0,1[, and n(x) = bloga(x)c. The binary operation Th⊕ion]0,1]given by
is a t-norm without zero divisors. In addition,Th⊕iis left-continuous (resp. strictly increasing on ]0,1]2) if and only if so doesT, andTh⊕i
]a,1] is order-isomorphic toT.
Remark 5.12 It is clear from the recursive description ofThki(see eq. (5.4)) that consecutive applications of Corollary 5.11 can result in all the t-norms, which can be generated by Theorem 5.8. Moreover, we see thatTh⊕k,...,⊕1i= Th⊕k,...,⊕i+1i
h⊕i,...,⊕1i