The most simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras are obtained, and the mutually independence of axioms is proved. We introduce the notions of filters and normal filters in pseudo-weak-R0 algebras. The structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak-R0 algebras.
1. Introduction
In recent years, the study of logic algebras and their noncommutative generalization—pseudo-logic algebras—has become of greater focus in the field of logic. BCK and BCI algebras were introduced by Imai and Iseki [1] and have been extensively investigated by many researchers. Georgescu and Iorgulescu [2] introduced the notion of a pseudo-BCK algebra as a noncommutative generalization of a BCK-algebra. Liu et al. [3] investigated the theory of pseudo-BCK algebras. MV-algebras were introduced by Chang in [4] as an algebraic tool to study the infinitely valued logic of Lukasiewicz. Georgescu and Iorgulescu [5] introduced pseudo MV-algebras which is a noncommutative generalization of MV-algebras. The notion of BL-algebras was introduced by Hajek [6] as the algebraic structures for his Basic Logic. Georgescu and Iorgulescu [7] introduced the notion of pseudo-BL algebras by dropping commutative axioms in BL-algebras. di Nola et al. [8, 9], Zhang and Fan [10], and Zhan et al. [11] investigated in detail the theory of pseudo-BL algebras. MTL-algebras [12] are the algebraic structures for Esteva-Godo monoidal t-norm based logic, many-valued propositional calculus that formalizes the structure of the real unit interval [0,1], induced by a left-continuous t-norm. Flondor et al. [13] presented pseudo-MTL algebras as a noncommutative generalization of MTL-algebras. IMTL-algebras [12] are the algebraic counterpart for involutive monoidal t-norm logic, an extension of MTL-algebras. NM-algebras [12] are the algebraic counterpart for nilpotent minimum logic, an extension of IMTL-algebras. Iorgulescu [14] and Liu and zhang [15] introduced and studied the pseudo-IMTL algebras and pseudo-NM algebras. R0 algebras were introduced by Wang [16] as the algebraic structure for his formal deductive system L* of fuzzy propositional calculus. Weak-R0 algebras [16] are the generalization of R0 algebras. The research on R0 algebras has attracted more and more attention [17].
In [18], we introduced and studied the pseudo-weak-R0 algebras and pseudo-R0 algebras. They are noncommutative generalizations of the weak-R0 algebras and R0 algebras, respectively. Some properties, the noncommutative forms of the properties in weak-R0 algebras and R0 algebras, were investigated. We showed that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras, and pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras.
Based on these results, in this paper, our study focused on the axioms independence and filter theory in pseudo-weak-R0 algebras and pseudo-R0 algebras. The most simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras are obtained, and the mutually independence of axioms is proved. The notions of filters and normal filters in pseudo-weak-R0 algebras are introduced. The structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak-R0 algebras.
2. Preliminaries
We recall some definitions and results which will be used in the sequel.
Definition 1 (see [12]).
An IMTL (involutive MTL) algebra is a structure (A,∨,∧,⊙,→,0,1) of type (2,2,2,2,0,0) such that for all x,y,z∈A:
(A,∨,∧,0,1) is a bounded lattice,
(A,⊙,1) is a monoid,
x⊙y≤z if and only if x≤y→z,
(x→y)∨(y→x)=1,
x--=x,
where x-=x→0.
An NM (nilpotent minimum) algebra is an IMTL algebra satisfying the following condition:
(x⊙y)-∨((x∧y)→(x⊙y))=1.
Definition 2 (see [14, 15]).
A pseudo-IMTL (pseudo-involutive MTL) algebra is a structure (A,∨,∧,⊙,→,⇝,0,1) of type (2,2,2,2,2,0,0) such that for all x,y,z∈A:
(A,∨,∧,0,1) is a bounded lattice,
(A,⊙,1) is a monoid,
x⊙y≤z if and only if x≤y→z if and only if y≤x⇝z,
(x→y)∨(y→x)=(x⇝y)∨(y⇝x)=1,
x~-=x-~=x,
where x-=x→0 and x~=x⇝0.
A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying the following condition:
(x⊙y)-∨((x∧y)→(x⊙y))=(x⊙y)~∨((x∧y)⇝(x⊙y))=1.
Definition 3 (see [16, 19]).
Let M be a (¬,∧,∨,→)-type algebra, where ¬ is a unary operation and ∧, ∨, and → are binary operations. If there is a partial ordering ≤ on M, such that (M,≤) is a bounded distributive lattice, ∧ and ∨ are infimum and supremum operations with respect to ≤, ¬ is an order-reversing involution with respect to ≤, and the following conditions hold for any a,b,c∈M
¬a→¬b=b→a,
1→a=a, a→a=1,
b→c≤(a→b)→(a→c),
a→(b→c)=b→(a→c),
a→(b∨c)=(a→b)∨(a→c), a→(b∧c)=(a→b)∧(a→c),
where 1 is the largest element of M, and then we call M a weak-R0 algebra.
An R0 algebra M is a weak-R0 algebra satisfying the additional condition as follows:
(a→b)∨((a→b)→(¬a∨b))=1.
Definition 4 (see [18]).
A pseudo-weak-R0 algebra is a structure(A,∧,∨,→,⇝,-,~,0,1) such that (A,∧,∨,0,1) is a bounded distributive lattice, - and ~ are order-reversing pseudo-involution (i.e., if x≤y, then y-≤x- and y~≤x~; x~-=x-~=x), and the following axioms hold for any x,y,z∈A:
A pseudo-R0 algebra A is a pseudo-weak-R0 algebra satisfying the additional axiom as follows:
(x→y)∨((x→y)⇝(x-∨y))=(x⇝y)∨((x⇝y)→(x~∨y))=1.
In [18], we also have another simplified definition.
Definition 5 (see [18]).
A pseudo-weak-R0 algebra is a structure (A,∧,∨,→,⇝,-,~,0,1) satisfying
(A,∧,∨,0,1) is a bounded lattice,
if x≤y, then y-≤x- and y~≤x~,
x~-=x-~=x,
x→y=y-⇝x-, x⇝y=y~→x~,
1→x=1⇝x=x,
x→y≤(z→x)→(z→y), x⇝y≤(z⇝x)⇝(z⇝y),
x→(y⇝z)=y⇝(x→z),
x→(y∨z)=(x→y)∨(x→z), x⇝(y∨z)=(x⇝y)∨(x⇝z).
A pseudo-R0 algebra A is a pseudo-weak-R0 algebra satisfying the additional axiom as follows:
(x→y)∨((x→y)⇝(x-∨y))=(x⇝y)∨((x⇝y)→(x~∨y))=1.
Proposition 6 (see [18]).
In a pseudo-weak-R0 algebra, the following properties hold:
0~=0-=1, 1~=1-=0,
x-=x→0, x~=x⇝0,
x⇝x=x→x=1,
x≤y if and only if x⇝y=1 if and only if x→y=1,
(⋀i∈Ixi)~=⋁i∈Ixi~, (⋀i∈Ixi)-=⋁i∈Ixi-, whenever the arbitrary meets and unions exist,
(⋁i∈Ixi)~=⋀i∈Ixi~, (⋁i∈Ixi)-=⋀i∈Ixi-, whenever the arbitrary meets and unions exist,
if x≤y, then z⇝x≤z⇝y and z→x≤z→y,
if x≤y, then y⇝z≤x⇝z and y→z≤x→z,
(x∧y)⇝z=(x⇝z)∨(y⇝z), (x∧y)→z=(x→z)∨(y→z),
x⇝(y∧z)=(x⇝y)∧(x⇝z), x→(y∧z)=(x→y)∧(x→z),
(x∨y)⇝z=(x⇝z)∧(y⇝z), (x∨y)→z=(x→z)∧(y→z),
x~∨y≤x⇝y, x-∨y≤x→y,
x⇝y≤(y⇝z)→(x⇝z), x→y≤(y→z)⇝(x→z),
(A,∧,∨,0,1) is a bounded distributive lattice,
x⇝y≤x∨z⇝y∨z, x→y≤x∨z→y∨z,
x⇝y≤x∧z⇝y∧z, x→y≤x∧z→y∧z,
(x⇝y)≤(x⇝z)∨(z⇝y), (x→y)≤(x→z)∨(z→y),
(x⇝y)∨(y⇝x)=(x→y)∨(y→x)=1,
x≤(x→y)⇝y, x≤(x⇝y)→y,
x→y=((x→y)⇝y)→y, x⇝y=((x⇝y)→y)⇝y,
x→(y→x)=x⇝(y⇝x)=x⇝(y→x)=x→(y⇝x)=1,
x-→(x→y)=x~⇝(x⇝y)=x-⇝(x→y)=x~→(x⇝y)=1,
y≤(x⇝y)∧(x→y),
x∨y=((x⇝y)→y)∧((y⇝x)→x)=((x→y)⇝y)∧((y→x)⇝x),
(x∨y)→x=y→x, (x∨y)⇝x=y⇝x,
x→(x∧y)=x→y, x⇝(x∧y)=x⇝y,
x≤y- if and only if y≤x~,
x→y~=y⇝x-, x⇝y-=y→x~,
(x→y-)~=(y⇝x~)-.
In a pseudo-weak-R0 algebra (pseudo-R0 algebra) A, we define a binary operation ⊙ as follows, for any x,y∈A:
(1)x⊙y=(x⟶y-)~=(y⇝x~)-.
Proposition 7 (see [18]).
In a pseudo-weak-R0 algebra, the following properties hold:
x→y=(x⊙y~)-, x⇝y=(y-⊙x)~,
(x⊙y)⊙z=x⊙(y⊙z),
1⊙x=x⊙1=x,
x⊙y≤z if and only if x≤y→z if and only if y≤x⇝z,
x⊙(x⇝y)≤y≤x⇝(x⊙y), (x→y)⊙x≤y≤x→(y⊙x),
x⊙(x⇝y)≤x≤y⇝(y⊙x), (x→y)⊙x≤x≤y→(x⊙y),
if x≤y, then x⊙z≤y⊙z and z⊙x≤z⊙y,
x⊙(x⇝y)≤x∧y, (x→y)⊙x≤x∧y,
x⊙0=0⊙x=0,
x⊙(⋁i∈Ixi)=⋁i∈I(x⊙xi), (⋁i∈Ixi)⊙x=⋁i∈I(xi⊙x), whenever the arbitrary unions exist,
(x⊙y)→z=x→(y→z), (y⊙x)⇝z=x⇝(y⇝z),
y⇝(⋀i∈Ixi)=⋀i∈I(y⇝xi), y→(⋀i∈Ixi)=⋀i∈I(y→xi), whenever the arbitrary meets exist,
(⋁i∈Ixi)⇝y=⋀i∈I(xi⇝y), (⋁i∈Ixi)→y=⋀i∈I(xi→y), whenever the arbitrary unions and meets exist,
x⊙x~=x-⊙x=0,
x⊙y≤x∧y≤x,y,
x∨(y⊙z)≥(x∨y)⊙(x∨z),
x→y≤(x⊙z)→(y⊙z), x⇝y≤(z⊙x)⇝(z⊙y),
x⊙(y→z)≤y→(x⊙z), (y⇝z)⊙x≤y⇝(z⊙x).
3. The Axioms Independence of Pseudo-Weak-R0 Algebras
We investigate the axioms independence of pseudo-R0 algebras and pseudo-weak-R0 algebras. Hence, we obtain most simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras.
Theorem 8.
A structure (A,∨,∧,→,⇝,-,~,0,1) is a pseudo-weak-R0 algebra if and only if it satisfies the following conditions:
(A,∧,∨,0,1) is a bounded lattice,
1~-=1-~=1, 0~-=0-~=0,
x→y=y-⇝x-, x⇝y=y~→x~,
1→x=1⇝x=x,
x→y≤(z→x)→(z→y), x⇝y≤(z⇝x)⇝(z⇝y),
x→(y∨z)=(x→y)∨(x→z), x⇝(y∨z)=(x⇝y)∨(x⇝z).
Proof.
Necessity is obvious. For sufficiency, it only needs to show axioms (pL2), (pL3), and (pR4) of Definition 5 hold. We first show the following three properties hold:
x→y≤(y→z)⇝(x→z), x⇝y≤(y⇝z)→(x⇝z),
x→x=x⇝x=1,
x≤y if and only if x→y=1 if and only if x⇝y=1.
In fact, by (pR1) and (pR3), we have x→y=y-⇝x-≤(z-⇝y-)⇝(z-⇝x-)=(y→z)⇝(x→z), x⇝y=y~→x~≤(z~→y~)→(z~→x~)=(y⇝z)→(x⇝z).
By (a) and (pR2), we have 1=1→1≤(1⇝x)→(1⇝x)=x→x, and so x→x=1. Similarly, x⇝x=1.
If x≤y, by (pR5) and (b), we have x→y=x→x∨y=(x→x)∨(x→y)=1. Conversely, if x→y=1, by (pR2) and (a), we have x=1⇝x≤(x→y)⇝(1→y)=1⇝y=y. Similarly, x≤y if and only if x⇝y=1.
(pL2): by (c) and (pR1), x≤y if and only if x→y=1 if and only if y-⇝x-=1 if and only if y-≤x-. Similarly, x≤y if and only if x⇝y=1 if and only if y~→x~=1 if and only if y~≤x~.
(pL3): since 0≤1-, by (pL2), 1-~≤0~. By (pL3′), 1≤0~; thus 1=0~ and 1-=0~-=0. Similarly, 1=0- and 1~=0.
By (pR2) and (pR1), x=1→x=x-⇝0, and so x-~=x-~-⇝0=1→x-~=x-⇝0=x. Hence, x-~=x. Similarly, we have x~-=x.
(pR4): by (pL2) and (pL3), it is easy to verify that pseudo-Kleene dual law holds:
(x∧y)~=x~∨y~, (x∨y)~=x~∧y~, (x∧y)-=x-∨y-, and (x∨y)-=x-∧y-.
By (pR1), (pR5), and (d), x∧y→z=z-⇝(x∧y)-=z-⇝x-∨y-=(z-⇝x-)∨(z-⇝y-)=(x→z)∨(y→z). Similarly, we have x∧y⇝z=(x⇝z)∨(y⇝z).
If x≤y, then y→z≤(x→z)∨(y→z)=x∧y→z=x→z and y⇝z≤(x⇝z)∨(y⇝z)=x∧y⇝z=x⇝z.
Now we prove that (pR4) holds. Since x=1→x≤(x→z)⇝(1→z)=(x→z)⇝z, x→(y⇝z)≥((x→z)⇝z)→(y⇝z)≥y⇝(x→z). Hence, x→(y⇝z)=y⇝(x→z).
Corollary 9.
A structure (A,∨,∧,→,⇝,-,~,0,1) is a pseudo-R0 algebra if and only if it satisfies (pL1), (pL3′), (pR1), (pR2), (pR3), (pR5), and
(x→y)∨((x→y)⇝(x-∨y))=(x⇝y)∨((x⇝y)→(x~∨y))=1.
According to Theorem 8 and Corollary 9, one obtains most simplified definitions of pseudo-weak-R0 algebras and pseudo-R0 algebras, as the axiom systems are mutually independence (see Theorem 11).
Definition 10.
A pseudo-weak-R0 algebra is a structure (A,∧,∨,→,⇝,-,~,0,1) such that (A,∧,∨,0,1) is a bounded lattice and (1~)-=(1-)~=1 and (0~)-=(0-)~=0, satisfying the following axioms:
x→y=y-⇝x-, x⇝y=y~→x~,
1→x=1⇝x=x,
x→y≤(z→x)→(z→y), x⇝y≤(z⇝x)⇝(z⇝y),
x→(y∨z)=(x→y)∨(x→z), x⇝(y∨z)=(x⇝y)∨(x⇝z).
A pseudo-R0 algebra A is a pseudo-weak-R0 algebra satisfying the additional axiom as follows:
(x→y)∨((x→y)⇝(x-∨y))=(x⇝y)∨((x⇝y)→(x~∨y))=1.
Theorem 11.
The five axioms of Definition 10 are mutually independent.
Proof.
Let A=[0,1], x∨y=max{x,y}, x∧y=min{x,y}, x-=1-x, and x~=1-x2. Then A is a bounded lattice satisfying x-~=x~-=x for any x∈A.
Define operations → and ⇝ as pseudo-Godel implication on A as follows:
(2)x⟶y=x⇝y={1,x≤y,y,otherwise.
Then A satisfies (P2)–(P5), but not (P1): 1→0.5=0.5, but 0.5-⇝1-=0.5⇝0=0.
Define operations → and ⇝ on A as follows:
(3)x⟶y=x⇝y=1.
Clearly, A satisfies (P1) and (P3)–(P5), but not (P2): 1→0.5=1⇝0.5=1≠0.5.
Define operations → and ⇝ on A as follows:
(4)x⟶y={x-∨y,x=1ory=0,1,otherwise,x⇝y={x~∨y,x=1ory=0,1,otherwise.
Then A satisfies (P1)-(P2) and (P4)-(P5), but not (P3). In fact, let x=0.64, y=0.1, and z=0, then y→z=y-∨z=y-=1-y=0.9≈0.95,(x→y)→(x→z)=1→x-∨z=1→x-=x-=0.36=0.6.
Define operations → and ⇝ as pseudo-Lukasiewicz implication on A as follows:
(5)x⟶y=1∧(x-+y),x⇝y=1∧(x~+y).
Then A satisfies (P1)–(P4), but not (P5): (0.19→0.07)∨[(0.19→0.07)⇝(0.19-∨0.07)]=0.97∨(0.97⇝0.9)≤0.97∨0.96=0.97<1.
Suppose that A is a bounded lattice given by Figure 1.
The operations -, ~, →, and ⇝ on A are defined by the following:
(6)x0fcdabe1x-=x~1ebadcf0x⟶y=x⇝y={1,x≤y,x-∨y=x~∨y,otherwise.
Then A satisfies (P1)–(P3) and (P5), but not (P4): (a→b)∨(a→c)=(a-∨b)∨(a-∨c)=(d∨b)∨(d∨c)=b∨a=e, but a→b∨c=a→e=1.
4. Filters and Normal Filters of Pseudo-Weak-R0 Algebras
We introduce the notions of filters and normal filters in pseudo-weak-R0 algebras and investigate the structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras.
Definition 12.
A nonempty subset F of a pseudo-weak-R0 algebra A is said to be a filter of A if it satisfies
x,y∈F⇒x⊙y∈F,
x∈F, x≤y⇒y∈F.
Proposition 13.
For a subset F of a pseudo-weak-R0 algebra A, the following are equivalent:
F is a filter,
1∈F and x,x⇝y∈F⇒y∈F,
1∈F and x,x→y∈F⇒y∈F.
Proof.
(i)⇒(ii). By (F2), we have 1∈F. By (F1), x,x⇝y∈F⇒x⊙(x⇝y)∈F. By (38) and (F2), x∧y∈F, and so y∈F.
(ii)⇒(iii). If x,x→y∈F, by (19), x≤(x→y)⇝y. By (4), x⇝((x→y)⇝y)=1∈F. By (ii), y∈F.
(iii)⇒(i). If x∈F, x≤y, then x→y=1∈F, so y∈F; that is, (F2) holds; if x,y∈F, by (41), x→(y→(x⊙y))=(x⊙y)→(x⊙y)=1∈F, and so x⊙y∈F, which means (F1) holds.
Clearly, {1} and A are both filters of a pseudo-weak-R0 algebra A.
Proposition 14.
For a subset F of a pseudo-weak-R0 algebra A, the following are equivalent:
F is a filter,
x,y∈F, y≤x→z⇒z∈F,
x,y∈F, y≤x⇝z⇒z∈F.
Proof.
(i)⇔(ii). If x,y∈F, y≤x→z, by (F2) and Proposition 13 (iii), z∈F. Conversely, if x∈F, by x≤x→1, we have 1∈F; suppose that x,x→y∈F, by x→y≤x→y, we have y∈F. By Proposition 13 (iii), F is a filter.
(i)⇔(iii). Similarly.
Next, we consider filter generated by a set. It is easy to verify that the intersection of filters of A is also a filter. If S⊆A, the least filter containing S; that is, the intersection of all filters of A containing S is called the filter generated by S and denoted by [S). If S={a}, [{a}) is written [a). Clearly
(7)[S)=∩{T∣S⊆T⊆A,TisafilterofA}.
Theorem 15.
Let A be a pseudo-weak-R0 algebra and let S be a nonempty subset of A. Then
(8)[S)={x∈A∣therearen≥1,a1,a2,…,an∈S,ksuchthata1⊙⋯⊙an≤x}={x∈A∣therearen≥1,a1,a2,…,an∈S,ksuchthatan⇝(⋯⇝(a1⇝x)⋯)=1}={x∈A∣therearen≥1,a1,a2,…,an∈S,ksuchthatan⟶(⋯⟶(a1⟶x)⋯)=1}.
Proof.
Only prove the first equality. Using (34) to the first equality, we can get the rest of the two equalities. Let B denote the right side of the first equality. If x,y∈B, then there are a1,a2,…,an,b1,b2,…,bm∈S such that a1⊙⋯⊙an≤x and b1⊙⋯⊙bm≤y. By (37), a1⊙⋯⊙an⊙b1⊙⋯⊙bm≤x⊙y, so x⊙y∈B. If x∈B and x≤y, we have a1⊙⋯⊙an≤x≤y, so y∈B. Hence B is a filter. If C is a filter and S⊆C, for any x∈B, there are a1,a2,…,an∈S such that a1⊙⋯⊙an≤x. By (F2), x∈C, hence B⊆C.
For convenience, we shall write an:=a⊙⋯⊙a︷n and a0:=1; a⇝nx:=a⇝(⋯⇝(a︷n⇝x)⋯) and a⇝0x:=x; a→nx:=a→(⋯→(a︷n→x)⋯) and a→0x:=x.
Corollary 16.
If A is a pseudo-weak-R0 algebra and a∈A, then
(9)[a)={x∈A∣n≥1,an≤x}={x∈A∣n≥1,a⇝nx=1}={x∈A∣n≥1,a→nx=1}.
Corollary 17.
Let F be a filter of a pseudo-weak-R0 algebra A and a∈A; then(10)[F∪{a})={x∈A∣(s1⊙an1)⊙⋯⊙(sm⊙anm)≤x,wherem≥1,n1,…,nm≥0,s1,…,sm∈F}.
Theorem 18.
Let F be a filter of a pseudo-weak-R0 algebra A and a,b∈A; then
(11)[F∪{a})∩[F∪{b})=[F∪{a∨b}).
Proof.
Assume that x∈[F∪{a})∩[F∪{b}), by Corollary 17, there are n1,…,nm,l1,…,lk≥0,s1,…,sm,t1,…,tk∈F such that
(12)(s1⊙an1)⊙⋯⊙(sm⊙anm)≤x,(t1⊙bl1)⊙⋯⊙(tk⊙blk)≤x.
Put p=s1⊙⋯⊙sm⊙t1⊙⋯⊙tk and q=max{n1,…,nm,l1,…,lk}, and then
(13)(p⊙aq)m≤x,(p⊙bq)k≤x.
Thus, by (46) x≥(p⊙aq)m∨(p⊙bq)k≥((p⊙aq)m∨(p⊙bq))k≥((p⊙aq)∨(p⊙bq))mk=(p⊙(aq∨bq))mk≥(p⊙(a∨b)q2)mk. x∈[F∪{a∨b}). Hence [F∪{a})∩[F∪{b})⊆[F∪{a∨b}). Inverse contains is obvious.
Corollary 19.
Let F be a filter of a pseudo-weak-R0 algebra A and a,b∈A. If a∨b∈F, then
(14)[F∪{a})∩[F∪{b})=F.
Corollary 20.
Let A be a pseudo-weak-R0 algebra and a,b∈A; then [a)∩[b)=[a∨b).
Proof.
Taking F={1} in Theorem 18.
Next we introduce the notion of normal filters in a pseudo-weak-R0 algebra.
Definition 21.
A filter F of a pseudo-weak-R0 algebra A is called normal if x,y∈A, x→y∈F if and only if x⇝y∈F.
Proposition 22.
Let F be a normal filter of a pseudo-weak-R0 algebra A. Then there is s∈F such that b⊙s≤c if and only if there is t∈F such that t⊙b≤c.
Proof.
If there is s∈F such that b⊙s≤c, by (34), s≤b⇝c. By s∈F, we have b⇝c∈F, and so b→c∈F. Put b→c=t∈F,and then t⊙b≤c. Converse is similar.
Theorem 23.
If F is a normal filter of a pseudo-weak-R0 algebra A and a∈A, then(15)[F∪{a})={x∈A∣thereares∈F,n≥0,suchthats⊙an≤x}={x∈A∣thereares∈F,n≥0,suchthatan⊙s≤x}.
Proof.
We show the first equality. By Corollary 17,
(16)[F∪{a})={x∈A∣(s1⊙an1)⊙⋯⊙(sm⊙anm)≤x,m≥1,n1,…,nm≥0s1,…,sm∈F}.
Since
(17)(s1⊙an1)⊙(s2⊙an2)⊙⋯⊙(sm⊙anm)≤x,
by (34),
(18)(s1⊙an1)⊙s2≤(an2⊙(s3⊙an3)⊙⋯⊙(sm⊙anm))⟶x,
by Proposition 22, there is t2∈F such that
(19)t2⊙(s1⊙an1)≤(an2⊙(s3⊙an3)⊙⋯⊙(sm⊙anm))⟶x,
and so
(20)(t2⊙s1)⊙an1+n2⊙(s3⊙an3)⊙⋯⊙(sm⊙anm)≤x.
Repeating the above steps, there are t2,…,tm∈F such that
(21)(tm⊙⋯⊙t2⊙s1)⊙an1+⋯+nm≤x.
Let s=tm+⋯+t2+s1∈F and n=n1+⋯+nm, we have s⊙an≤x. That is that the first equality holds.
By the first equality and Proposition 22, we can obtain the second equation.
Corollary 24.
If F is a normal filter of a pseudo-weak-R0 algebra A and a∈A, then(22)[F∪{a})={x∈A∣thereisn≥0,suchthata→nx∈F}={x∈A∣thereisn≥0,suchthata⇝nx∈F}.
Proof.
By Theorem 23,
(23)[F∪{a})={x∈A∣thereares∈F,n≥0,suchthats⊙an≤x}.
Since there is s∈F such that s⊙an≤x, if and only if there is s∈F such that s≤a→(⋯→(a︷n→x)⋯); that is, there is s∈F such that s≤a→nx, if and only if a→nx∈F. Thus, we prove the first equality.
Similarly, by
(24)[F∪{a})={x∈A∣thereares∈F,n≥0,suchthatan⊙s≤x}.
Since there is s∈F such that an⊙s≤x, if and only if there is s∈F such that s≤a⇝(⋯⇝(a︷n⇝x)⋯); that is, there is s∈F such that s≤a⇝nx, if and only if a⇝nx∈F, Thus, we have the second equality.
Corollary 25.
If F is a normal filter of a pseudo-weak-R0 algebra A and a∈A, then(25)[F∪{a})={x∈A∣thereisn≥0,suchthatan⟶x∈F}={x∈A∣thereisn≥0,suchthatan⇝x∈F}.
Proof.
There is s∈F such that s⊙an≤x, if and only if there is s∈F such that s≤an→x, if and only if an→x∈F. There is s∈F such that an⊙s≤x, if and only if there is s∈F such that s≤an⇝x, if and only if an⇝x∈F. By Theorem 23, Corollary 25 holds.
5. Conclusions
We obtained the most simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras and proved the mutually independence of axioms. We introduced the notions of filters and normal filters in pseudo-weak-R0 algebras and gave the structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras. These will be conducive to further study pseudo-weak-R0 algebras (pseudo-IMTL algebras) and pseudo-R0 algebras (pseudo-NM algebras). In the future, we will investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak-R0 algebras and pseudo-R0 algebras.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundations of China (61175055), the Fujian Province Natural Science Foundations of China (2013J01017), and the Fujian Province Key Project of Science and Technology of China (2011Y0049).
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