On the Axiomatic Presentation of Pseudo-Hoops
Rodica Ceterchi
Faculty of Mathematics and Computer Science
University of Bucharest
[email protected], [email protected]
Abstract
Pseudo-hoops are basically partially ordered monoids, not necessarily commutative, with residuations. The algebras of (non-commutative) fuzzy logic, pseudo-BL and pseudo-MV algebras, arise as
particular cases of pseudo-hoops. Following work by Bosbach, we give a characterization of pseudohoops with equations.
1
Introduction
The term ”pseudo-hoops” was coined by G. Georgescu, L. Leustean and V. Preoteasa in their seminal
paper [15]. The prefix ”pseudo-” stands for non-commutative or not necessarily commutative type of
algebra. It followed naturally after the introduction of pseudo-MV algebras ([12], [13]), pseudo-Wajsberg
algebras ([6],[7]), and pseudo-BL algebras ([14],[8],[9]). All the above (and also pseudo-BCK algebras [17])
are non-commutative generalizations of algebras for many-valued logics.
Pseudo-hoops are weaker structures, and pseudo-MV, pseudo-Wajsberg, and pseudo-BL algebras arise
as particular cases of them.
Pseudo-hoops are monoids endowed with orders. Moreover, the orders are canonical (actually inverse
canonical) – they are given by divisibility relations w.r.t. the monoid operation – and the orders have
residuals.
Definition 1.1 [15] An algebra (A, , 1, →, ;) of type (2, 0, 2, 2) is a pseudo-hoop if
(i) (A, , 1, →) is a left-complemented monoid.
(ii) (A, , 1, ;) is a right-complemented monoid.
(iii) ≤r =≤l (and we denote it by ≤).
The authors of [15] discovered that the structure was studied before: first, by Bosbach, in a series of
papers which includes [2] and [3]; next, by Buchi and Owens, in an unfortunately unpublished paper [5],
and next by Blok and Pigozzi, in [1], who mention [5]. In Bosbach’s papers, the structure was called hoop,
and no assumption was made about the commutativity of the monoid operation. Blok continued the study
begun in [1] with a series of papers co-authored with Ferreirim, and hoops were included in the class of
algebras for fuzzy logic by subsequent papers, but only with the additional axiom of commutativity of the
monoid operation.
The question naturally arises, whether these structures have an axiomatic presentation with equations,
as is the case with pseudo-MV, pseudo-Wajsberg, and pseudo-BL algebras.
The answer is positive, and is given by the following:
Theorem 1.2 ([15], Theorem 2.2) An algebra (A, , 1, →, ;) of type (2, 0, 2, 2) is a pseudo-hoop iff
(H1) a 1 = 1 a = a
(H2) a → a = a ; a = 1
(H3) (a b) → c = a → (b → c)
1
(H3’) (a b) ; c = b ; (a ; c)
(H4) (a → b) a = (b → a) b = a (a ; b) = b (b ; a).
This equational characterization of pseudo-hoops, follows from the next two Theorems, 1.3 and 1.4.
Theorem 1.3 The equational characterization of left-complemented monoids
An algebra (A, , 1, →) of type (2, 0, 2) is a left-complemented monoid if and only if the following conditions
are fulfilled:
(H1) a 1 = 1 a = a
(H2) a → a = 1
(H3) (a b) → c = a → (b → c)
(H4) (a → b) a = (b → a) b
The dual of Theorem 1.3 is
Theorem 1.4 The equational characterization of right-complemented monoids
An algebra (A, , 1, ;) of type (2, 0, 2) is a right-complemented monoid if and only if the following hold:
(H1) a 1 = 1 a = a
(H2’) a ; a = 1
(H3’) (a b) ; c = b ; (a ; c)
(H4’) a (a ; b) = b (b ; a)
Theorem 1.3 is Proposition 1.2 of [15], and Theorem 1.4 is Proposition 1.3 of [15]. For both of them,
the paper [15] proves one implication, namely that the definition implies equations (H), and quotes [2],
[3] and [5] for the other (hard) implication.
It is the purpose of this note to fill this gap. Section 2 is devoted to introducing the notions of leftand right-complemented monoids, and to the presentation of some of their first properties, which include
axioms (H), thus proving one implication of Theorems 1.3 and 1.4. We also stress here how the study of
semigroups with natural orders diverges from that of groups. Section 3 is dedicated to proving in detail
the other implications, following work by Bosbach in [2]. We prove actually a stronger result of Bosbach,
formulated here as Theorem 3.11. We feel that this update might be welcome, in view of the growing
interest in pseudo-hoops as algebras for non-commutative fuzzy logic, as shown by recent papers, e.g. [10],
[11] and [4].
2
Left- and Right-complemented Monoids
Definition 2.1 We call (A, , 1, ≤) a partially ordered monoid (po-monoid for short) if:
(i) (A, , 1) is a monoid, i.e. is associative and x 1 = 1 x = x,
(ii) (A, ≤) is a partially ordered set (a poset),
(iii) the monoid operation is increasing in both arguments, i.e.
is left-increasing (increasing in the first argument): a ≤ b ⇒ a c ≤ b c
is right-increasing (increasing in the second argument): a ≤ b ⇒ c a ≤ c b
If is left-increasing we call (A, , 1, ≤) a left po-monoid, and if it is right-increasing we call (A, , 1, ≤)
a right po-monoid.
Consider (A, , 1, ≤) a left po-monoid. If the greatest element of the set {x ∈ A | x a ≤ b} exists, it
is denoted a → b, and is called the left residual of .
If (A, , 1, ≤) is a right po-monoid, and if the greatest element of the set {x ∈ A | a x ≤ b} exists, it
is denoted a ; b, and is called the right residual of .
If they exist for every a, b ∈ A the residuals are characterized by the conditions
xa≤b⇔x≤a→b
2
ax≤b⇔x≤a;b
respectively, and the monoids are called left residuated monoid, respectively right residuated monoid.
Note that if operation is commutative, the notions of left-increasing and right-increasing coincide,
and so do the left and right residuals, if they exist. Actually, we have:
Proposition 2.2 Let (A, , 1, ≤) be a po-monoid, with left and right residuals → and ;. Then is
commutative iff →=;.
2.1
Canonically Ordered Semigroups
In any semigroup (A, ), i.e. in the presence of an associative binary operation, some natural pre-orders
arise, given by divisibility. If is not commutative, we will have to distinguish between left and right
divisors as Example 2.4 shows.
Furthermore, divisors can be smaller than what they divide, (as in Z+ ), in which case we will use the
term canonically (pre-)ordered or naturally (pre-)ordered. Divisors can be greater than what they divide,
(as in Z− ), in which case we will use the term inverse canonically/naturally (pre-)ordered.
We can thus define
the right divisibility relation, ≤0r , to be the relation (a is a right divisor of b):
a ≤0r b ⇔ there exists x, such that b = x a.
the inverse right divisibility relation, ≤r , to be the relation (b is a right divisor of a):
a ≤r b ⇔ there exists x, such that a = x b.
Analogous definitions follow for left divisibility, relations ≤0l and ≤l .
All four relations are pre-orders, i.e. they are reflexive and transitive (we will prove this in the next
subsection), but they are not always antisymmetric.
Definition 2.3 A semigroup (A, ) will be called (inverse) canonically ordered iff the (inverse) canonical
pre-order is an order.
Example 2.4 (M ∗ , ·, λ) the free monoid generated by M , where · is catenation, and λ is the empty word.
Then m1 ≤0l m2 iff m2 = m1 · x i.e. m1 is a prefix of m2 . On the other hand, m1 ≤0r m2 iff m2 = y · m1
i.e. m1 is a suffix of m2 . The two orders do not coincide.
Example 2.5 (R+ , +, 0, ≤) is a canonically ordered monoid.
Example 2.6 (R− , +, 0, ≤) is an inverse canonically ordered monoid.
Example 2.7 (Z+ , +, 0, ≤) is a canonically ordered monoid.
Example 2.8 (Z− , +, 0, ≤) is an inverse canonically ordered monoid.
Example 2.9 (R, +, 0, ≤) is not canonically ordered, it is a group.
Counter-example 2.9 is justified by the following result from [16]:
Proposition 2.10 A monoid (A, , 1) cannot be simultaneously a group and canonically ordered.
Proof: Assume that (A, , 1, −1 ) is a group, and that it is also canonically ordered. Take two elements,
a 6= b. Then, there exists c = b−1 a, such that a = b c, thus b ≤0l a. There also exists d = a−1 b,
such that b = a d, thus a ≤0l b. Since ≤0l is an order, it follows a = b.
2
3
2.2
Left-complemented Monoids
Let (A, , 1) be a monoid. We define the inverse right divisibility relation, ≤r , to be the relation:
a ≤r b ⇔ there exists x, such that a = x b.
Lemma 2.11 The inverse right divisibility relation has the following properties:
1) a ≤r a (≤r reflexive)
2) a ≤r b ≤r c ⇒ a ≤r c (≤r transitive)
3) a ≤r 1 (1 is ”greatest element”)
4) is left-increasing w. r. t. ≤r .
Proof:
1) a ≤r a because a = 1 a.
2) a ≤r b ≤r c implies a = x b = x (y c) = (x y) c, thus a ≤r c, by associativity of .
3) a ≤r 1 because a = a 1.
4) Let a ≤r b, so a = x b. It follows a c = (x b) c = x (b c), thus a c ≤r b c.
2
Note that ≤r is a pre-order, and might not always be an order relation, i.e. antisymmetric. In the
following we will consider structures for which (A, , 1, ≤r ) is a left po-monoid, and furthermore, which
have (left) residuals.
Definition 2.12 An algebra (A, , 1, →) is called a left-complemented monoid if
(a) (A, , 1, ≤r ) is a left po-monoid; (i.e. (A, , 1) is a monoid with ≤r antisymmetric)
(b) has → as left residual, i.e. we have the left-residuation condition
(R)
x a ≤r b ⇔ x ≤r a → b.
Proposition 2.13 Let (A, , 1, →) be a left-complemented monoid. Then the following are true for every
a, b, c ∈ A:
(1) a → a = 1
(H2)
(2) a → 1 = 1
(3) (a b) → c = a → (b → c)
(H3)
Proof:
(1) From a ≤r a we have 1 a ≤r a, and from (R), 1 ≤r a → a. But we also have a → a ≤r 1 so
a → a = 1.
(2) From a ≤r 1 we have 1 a ≤r 1, and from (R), 1 ≤r a → 1, thus a → 1 = 1.
(3) x ≤r (a b) → c iff, by (R), x (a b) ≤r c iff, by associativity of , (x a) b ≤r c, iff, by (R),
(x a) ≤r b → c, iff, by (R), x ≤r a → (b → c).
2
Lemma 2.14 Let (A, , 1, →) be a left-complemented monoid. Then
(a → b) a = inf≤r {a, b}
4
Proof:
(a → b) a ≤r b iff, by (R), a → b ≤r a → b, which is true by reflexivity of ≤r . From a → b ≤r 1 and the
fact that is left-increasing we have (a → b) a ≤r 1 a = a. Thus (a → b) a is a minorant for {a, b}.
Take a z such that z ≤r a and z ≤r b. It follows that z = x a and z = y b. Thus x a = y b ≤r b,
from which by (R) follows x ≤r a → b, from which, by left-increasing, x a ≤r (a → b) a. But this
last relation is precisely z ≤r (a → b) a.
2
Corollary 2.15 Let (A, , 1, →) be a left-complemented monoid. Then for every a, b ∈ A we have:
(4) (a → b) a = (b → a) b.
(H4)
The relations (H1) − (H4) have been shown to hold in a left-complemented monoid, thus completing
the proof of one implication of Theorem 1.3. The reverse implication is the subject of the next section.
Let us note that the order can be retrieved via the residual.
Lemma 2.16 Let (A, , 1, →) be a left-complemented monoid. Then
a ≤r b ⇔ a → b = 1.
Proof:
a ≤r b ⇔ a = x b ⇒ a → b = (x b) → b = x → (b → b) = x → 1 = 1, by (H3), (H2) and (2).
From a → b = 1 and (H4) (a → b) a = (b → a) b we get a = 1 a = (b → a) b.
2.3
2
Right-complemented Monoids
Analogously, we define the notion of right-complemented monoid.
Let ≤l be the inverse left divisibility relation defined by
a ≤l b ⇔ there exists x, such that a = b x.
The proofs of the following results are similar to the ones for the left case, so we omitt them.
Lemma 2.17 The inverse left divisibility relation has the following properties:
1) a ≤l a (≤l reflexive)
2) a ≤l b ≤l c ⇒ a ≤l c (≤l transitive)
3) a ≤l 1 (1 is ”greatest element”)
4) is right-increasing w. r. t. ≤l .
Definition 2.18 An algebra (A, , 1, ;) is called a right-complemented monoid if
(a) (A, , 1, ≤l ) is a right po-monoid; (i.e. (A, , 1) is a monoid with ≤l antisymmetric)
(b) has ; as right residual, i.e. we have the right-residuation condition
(R0 )
a x ≤l b ⇔ x ≤l a ; b.
Proposition 2.19 Let (A, , 1, ;) be a right-complemented monoid. Then the following are true for
every a, b, c ∈ A:
(1’) a ; a = 1
(H2’)
(2’) a ; 1 = 1
(3’) (a b) ; c = b ; (a ; c)
(H3’)
Lemma 2.20 Let (A, , 1, ;) be a right-complemented monoid. Then
5
a (a ; b) = inf≤l {a, b}
Corollary 2.21 Let (A, , 1, ;) be a right-complemented monoid. Then for every a, b ∈ A we have:
(4’) a (a ; b) = b (b ; a).
(H4’)
This completes the proof of one implication of Theorem 1.4.
Again, the order can be retrieved via the residual.
Lemma 2.22 Let (A, , 1, ;) be a right-complemented monoid. Then
a ≤l b ⇔ a ; b = 1.
3
An equational characterization of complemented monoids
3.1
Equational presentation of right-complemented monoids
We present in detail, following [2], the equational characterization of right-complemented monoids. We
will prove actually a stronger result.
Consider an algebra (A, , ;) of type (2, 2), satisfying the following axioms:
(H1) a (a ; b) = b (b ; a)
(H2) (a b) ; c = b ; (a ; c)
(H3) a (b ; b) = a
We will prove that these three axioms completely characterize the right-complemented monoids. Note
that (A, ) is not supposed to be a monoid, nor even a semigroup. As we will see below, associativity of
and the existence of the neutral element can be derived from the above axioms.
Theorem 3.1 Let (A, , ;) be an algebra of type (2, 2), satisfying the axioms:
(H1) a (a ; b) = b (b ; a)
(H2) (a b) ; c = b ; (a ; c)
(H3) a (b ; b) = a
Then, there exists a constant 1 ∈ A such that (A, , ;, 1) is a right-complemented monoid.
The proof will be split into a series of Lemmas which derive consequences of the axioms.
Lemma 3.2 We have the following consequences of axioms (H1), (H2) and (H3):
(1) a ; a = b ; b. This implies there exists a constant in A which we denote 1, and 1 = a ; a.
(2) (a b) c ; d = a (b c) ; d.
(3) (a b) c = a (b c). (Operation is associative.)
Proof:
(1) a ; a = (a ; a) (b ; b)
by (H3)
= (a ; a) [b (a ; a) ; b]
by (H3)
= (a ; a) [(a ; a) ; (b ; b)]
by (H2)
= (b ; b) [(b ; b) ; (a ; a)]
by (H1)
6
= (b ; b) [a (b ; b) ; a]
by (H2)
= (b ; b) (a ; a)
by (H3)
= (b ; b)
by (H3)
2
= 1.
(2) (a b) c ; d = c ; ((a b) ; d)
by (H2)
= c ; (b ; (a ; d))
by (H2)
= (b c) ; (a ; d)
by (H2)
= a (b c) ; d
by (H2)
2
(3) (a b) c = [(a b) c] [a (b c) ; a (b c)]
by (H3)
= [(a b) c] [(a b) c ; a (b c)]
by (2)
= [a (b c)] [a (b c) ; (a b) c]
by (H1)
= [a (b c)] [(a b) c ; (a b) c]
by (2)
= [a (b c)].
by (H3)
2
Corollary 3.3 If (A, , ;), an algebra of type (2, 2), satisfies (H1), (H2) and (H3), then (A, , 1) is a
semigroup with right unit.
The fact that 1 is a right unit follows immediately from (1) and (H3). The next result leads us step
by step to the conclusion that 1 is also a left unit.
Lemma 3.4 We have the following consequences of axioms (H1), (H2) and (H3):
(4) (1 ; a) ; 1 = 1.
(5) 1 ; a = 1 (1 ; a).
(6) (1 ; a) ; a = 1.
(7) 1 ; a = 1 a (a ; 1 a).
(8) 1 ; (1 a) = 1 a.
(9) 1 a = 1 ; a.
(10) 1 a = a.
Proof:
(4) (1 ; a) ; 1 = (1 ; a) ; (1 ; 1)
by (1)
= (1 (1 ; a)) ; 1
by (H2)
= (a (a ; 1)) ; 1
by (H1)
= (a ; 1) ; (a ; 1)
by (H2)
= 1.
by (1)
2
7
(5) 1 ; a = (1 ; a) 1
by (H3) and (1)
= (1 ; a) [(1 ; a) ; 1]
by (4)
= 1 [1 ; (1 ; a)]
by (H1)
= 1 [(1 1) ; a]
by (H2)
= 1 (1 ; a).
2
(6) (1 ; a) ; a = 1 (1 ; a) ; a
by (5)
= (1 ; a) ; (1 ; a)
by (H2)
=1
by (1)
2
(7) 1 ; a = (1 ; a) 1
by 1 right unit
= (1 ; a) [(1 ; a) ; a]
by (6)
= a [a ; (1 ; a)]
by (H1)
= a [(1 a) ; a]
by (H2)
= 1 a [(1 a) ; a]
by (5) and (3)
= a [a ; (1 a)]
by (H1)
= 1 a [a ; (1 a)]
by (5)
2
(8) 1 ; (1 a) = 1 (1 a) [(1 a) ; 1 (1 a)]
= (1 a) [(1 a) ; (1 a)]
by (7)
by (3)
= (1 a)
by (H3)
2
(9) 1 a = (1 a) [(1 a) ; (1 a)]
by (H3)
= (1 a) [a ; (1 ; (1 a))]
by (H2)
= (1 a) [a ; (1 a)]
by (8)
=1;a
by (7)
2
(10) 1 a = 1 ; a
by (9)
= (1 ; a) 1
by (H3)
= (1 ; a) [(1 ; a) ; a]
by (6)
= a [a ; (1 ; a)]
by (H1)
= a [(1 a) ; a]
by (H2)
8
= a [(1 ; a) ; a]
by (9)
= a1
by (6)
=a
2
Corollary 3.5 If (A, , ;), an algebra of type (2, 2), satisfies (H1), (H2) and (H3), then (A, , 1) is a
monoid.
Relation (10) states that 1 is also a left unit, so we have a 1 = 1 a = a.
Lemma 3.6 We have the following consequences of axioms (H1), (H2) and (H3):
(11) 1 ; a = a.
(12) a ; 1 = 1.
(13) a x = b ⇔ b ; a = 1
(14) a x = b and b y = a ⇒ a = b.
(15) a x = b y ⇔ x ; (a ; b) = 1.
Proof:
(11) 1 ; a = 1 a = a, by (9) and (10).
2
(12) a ; 1 = (1 a) ; 1
by (10)
= (1 (1 ; a)) ; 1
by (11)
= a (a ; 1)) ; 1
by (H1)
= (a ; 1) ; (a ; 1)
by (H2)
=1
by (1)
2
(13) b = a x ⇒ b ; a = (a x) ; a
= x ; (a ; a)
by (H2)
=x;1=1
by (1) and (12)
b ; a = 1 ⇒ b = b 1 = b (b ; a)
= a (a ; b)
by (H1)
(14) a x = b ⇔ b ; a = 1, and b y = a ⇔ a ; b = 1, by (13). Next we have:
a = a 1 = a (a ; b) = b (b ; a) = b 1 = b, by (H1).
(15) a x = b y ⇒ x ; (a ; b) = (a x) ; b
2
by (H2)
= (b y) ; b = 1
by (13)
x ; (a ; b) = 1 ⇒ (a x) ; b = 1
by (H2)
⇒ax=by
by (13)
9
2
Proof of Theorem 3.1:
(A, ≤l ) is a po-set, because ≤l is antisymmetric by (14), and thus an order relation. We also have, by
(13), that a ≤l b iff a ; b = 1.
(A, , 1, ≤l ) is a right po-monoid, integral. is increasing in the second argument w.r.t. ≤l , and 1 is
greatest element, by associativity of and 1 neutral.
(A, , 1, ;, ≤l ) is right-residuated. This follows from the equivalences
a x ≤l b ⇔ a x = b y ⇔ x ; (a ; b) = 1 ⇔ x ≤l a ; b
which use the definition of ≤l , (15) and (13).
2
Theorem 3.1 concludes the proof of Theorem 1.4.
Note that we can prove that a (a ; b) = inf≤l {a, b} directly as a consequence of axioms (H1), (H2),
(H3).
Lemma 3.7 Let (A, , 1, ;) be an algebra satisfying (H1), (H2), (H3). Then a (a ; b) = inf≤l {a, b}.
Proof:
We prove first that we have relation
(16) a (a ; b) ; a = b (b ; a) ; b = 1.
(16) a (a ; b) ; a = (a ; b) ; (a ; a)
by (H2)
= (a ; b) ; 1
by (1)
=1
by (12)
= (b ; a) ; 1
by (12)
= (b ; a) ; (b ; b)
by (1)
= b (b ; a) ; b.
by (H2)
From(16) it follows that a (a ; b) is a minorant for a and b.
Now let c be such that c ≤l a, b, i.e. c ; a = c ; b = 1. We have:
(a ; c) ; (a ; b) = a (a ; c) ; b
by (H2)
= c (c ; a) ; b
by (H1)
= (c ; a) ; (c ; b)
by (H2)
= 1 ; 1 = 1.
Thus a ; c ≤l a ; b. From this, since is right-increasing, a (a ; c) ≤l a (a ; b). But
c = c 1 = c (c ; a) = a (a ; c), so we have the desired c ≤l a (a ; b).
2
10
3.2
Equational presentation of left-complemented monoids
Let us now present the equational presentation of left-complemented monoids. The proofs are similar to
the ones for the ”right” case, so we omitt them.
Theorem 3.8 Let (A, , →) be an algebra of type (2,2) satisfying the axioms:
(H1’) (a → b) a = (b → a) b
(H2’) (a b) → c = a → (b → c)
(H3’) (b → b) a = a
Then there exists a constant 1 ∈ A such that (A, , 1, →) is a left-complemented monoid.
The proof is based on the following consequences of the axioms, which are duals of the ones established
in the previous subsection.
Lemma 3.9 Let (A, , →) be an algebra of type (2,2) satisfying the axioms (H1’), (H2’) and (H3’). Then
the following relations hold:
(1’) a → a = b → b. This implies there exists a constant 1 ∈ A, and 1 = a → a.
(2’) (a b) c → d = a (b c) → d.
(3’) (a b) c = a (b c).
(4’) (1 → a) → 1 = 1.
(5’) 1 → a = (1 → a) 1.
(6’) (1 → a) → a = 1.
(7’) 1 → a = (a → a 1) a 1.
(8’) 1 → (a 1) = a 1.
(9’) a 1 = 1 → a.
(10’) a 1 = a.
(11’) 1 → a = a.
(12’) a → 1 = 1.
(13’) x a = b ⇔ b → a = 1.
(14’) x a = b and y b = a imply a = b.
(15’) x a = y b ⇔ x → (a → b) = 1.
From relations (10 ) − (150 ) the proof of Theorem 3.8 follows.
The following is the dual of Lemma 3.7.
Lemma 3.10 Let (A, , 1, ;) be an algebra satisfying (H1’), (H2’), (H3’).
inf≤r {a, b}.
3.3
Then (a → b) a =
A stronger axiomatic
By putting together Theorems 3.8 and 3.1 we obtain a stronger equational characterization of pseudohoops, as algebras of type (2, 2, 2), result due to Bosbach [2].
Theorem 3.11 Let (A, , →, ;) be an algebra of type (2, 2, 2), satisfying the following axioms:
(H1) (a → b) a = (b → a) b = a (a ; b) = b (b ; a)
(H2) (a b) ; c = b ; (a ; c)
(H2’) (a b) → c = a → (b → c)
(H3) a (b ; b) = (b → b) a = a.
Then, there exists a constant in A, 1 = a → a = a ; a, such that (A, , 1, →, ;) is a pseudo-hoop.
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