Pseudo-BCI algebras with derivations
Lavinia Corina Ciungu
Abstract.
In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate
their properties and we give a characterization of regular implicative derivations of type II.
We also define the notion of a d-invariant deductive system of a pseudo-BCI algebra A proving that
d is a regular derivation of type II if and only if every deductive system on A is d-invariant.
It is proved that a pseudo-BCI algebra is p-semisimple if and only if the only regular derivation of type II
is the identity map. Another main result consists of proving that the set of all implicative derivations
of a p-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition.
Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of
p-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations
are equal.
Keywords: pseudo-BCI algebra, commutative pseudo-BCI algebra, p-semisimple pseudo-BCI algebra, implicative derivation, regular derivation, isotone derivation, invariant deductive system, symmetric derivation
AMS Mathematics Subject Classification (2010): 06D35, 06F05, 03F50
1. Introduction
The notion of derivations from the analytic theory was introduced in 1957 by Posner ([41]) to a prime ring
(R,+,⋅) as a map d:R⟶R satisfying the conditions d(x+y)=d(x)+d(y) and
d(x⋅y)=d(x)⋅y+x⋅d(y), for all x,y∈R. Since the derivation proved to be helpful for studying
the properties of algebraic systems, this notion has been defined and studied by many authors for the cases of
lattices ([44], [23], [45], [46]) and algebras of fuzzy logic:
MV-algebras ([3]), [47], [22]), BCI-algebras ([30], [1], [2]), commutative residuated lattice ([25]), BCC-algebras ([42], [4]), BE-algebras ([32])), basic algebras ([34]) and pseudo-MV algebras ([43]).
The aim of this paper is to introduce the concept of derivations on pseudo-BCI algebras and to investigate their
properties. We define the type I and type II implicative derivations on pseudo-BCI algebras, we introduce the notion
of a regular implicative derivation and we give a characterization of regular implicative derivations of type II.
The notion of isotone implicative derivations is also defined, and it is proved that any regular implicative
derivation of type II is isotone.
For an implicative derivation d on a pseudo-BCK algebra A we define the notion of a d-invariant deductive
system of A proving that d is a regular implicative derivation of type II if and only if every deductive
system on A is d-invariant.
We investigate the particular case of implicative derivations on the p-semisimple pseudo-BCI algebras and
we prove that a pseudo-BCI algebra is p-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations
of a p-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition.
Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of
p-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations
are equal.
2. Preliminaries
Pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu in [21] as algebras
with ”two differences”, a left- and right-difference, and with a constant element [math] as the least element. Nowadays pseudo-BCK algebras are used in a dual form, with two implications, → and ⇝ and with one constant element 1, that is the greatest element. Thus such pseudo-BCK algebras are in the ”negative cone” and are also called ”left-ones”. Pseudo-BCK algebras were intensively studied in [27], [28], [26], [35], [9].
Pseudo-BCI algebras were defined by [11] as generalizations of pseudo-BCK algebra and BCI-algebras, and they
form an important tool for an algebraic axiomatization of implicational fragment of non-classical logic ([15]).
In this section we recall some basic notions and results regarding pseudo-BCI algebras from [11], [12]-[19], [7]-[8], [20].
Definition 2.1**.**
([8]) A pseudo-BCI algebra is a structure
A=(A,→,⇝,1) of type (2,2,0) satisfying the following axioms, for all x,y,z∈A:
(psBCI1) (x→y)⇝[(y→z)⇝(x→z)]=1;
(psBCI2) (x⇝y)→[(y⇝z)→(x⇝z)]=1;
(psBCI3) 1→x=x;
(psBCI4) 1⇝x=x;
(psBCI5) (x→y=1 and y→x=1) implies x=y.
It is proved in [8, Lemma 2.1] that x→y=1 iff x⇝y=1, so that axiom (psBCI5) is equivalent to the following axiom:
(psBCI5′) (x⇝y=1 and y⇝x=1) implies x=y.
Every pseudo-BCI algebra satisfying x→y=x⇝y for all x,y∈A is a BCI-algebra, and every pseudo-BCI algebra
satisfying x≤1 for all x∈A is a pseudo-BCK algebra.
A pseudo-BCI algebra is said to be proper if it is not a BCI-algebra and it is not a pseudo-BCK algebra.
In a pseudo-BCI algebra (A,→,⇝,1), one can define a binary relation ‘‘≤" by
x≤y iff x→y=1 iff x⇝y=1, for all x,y∈A.
We will refer to (A,→,⇝,1) by its universe A.
Lemma 2.2**.**
([20])* Let (A,→,⇝,1) be a pseudo-BCI algebra. Then the following hold for all x,y,z∈A:
(1) x→x=x⇝x=1;
(2) x≤(x→y)⇝y and x≤(x⇝y)→y;
(3) x→y=1 iff x⇝y=1;
(4) x≤y→z iff y≤x⇝z;
(5) x≤y implies y→z≤x→z and y⇝z≤x⇝z;
(6) x≤y implies z→x≤z→y and z⇝x≤z⇝y;
(7) x→y≤(z→x)→(z→y) and x⇝y≤(z⇝x)⇝(z⇝y);
(8) x→(y⇝z)=y⇝(x→z);
(9) if 1≤x, then x=1;
(10) if x≤y and y≤z, then x≤z;
(11) x→1=x⇝1;
(12) (x→y)→1=(x→1)⇝(y→1) and (x⇝y)⇝1=(x⇝1)→(y⇝1).*
For any pseudo-BCI algebra A the set K(A)={x∈A∣x≤1} is a subalgebra of A called the
pseudo-BCK part of A, since (K(A),→,⇝,1) is a pseudo-BCK algebra.
An element a of a pseudo-BCI algebra A is called an atom if a≤x implies x=a, for
all x∈A. Denote by At(A) the set of all atoms of A and it is proved in [14] that
At(A)={x∈A∣x=(x→1)→1}.
For any a∈A we denote V(a)={x∈X∣x≤a} and it is obvious that V(a)=∅, since
a≤a gives a∈V(a). If a∈At(A), then the set V(a) is called a branch of A determined by
the element a. According to [13] we have: (1) branches determined by different elements are disjoints;
(2) a pseudo-BCI algebra is a set-theoretic union of branches; (3) comparable elements are in the same branch;
(4) the elements x and y belong to the same branch if and only if x→y∈V(1) or equivalently
x⇝y∈V(1).
A pseudo-BCI algebra A is p-semisimple if x≤1 implies x=1, for all x∈A, that is K(A)={1}.
Proposition 2.3**.**
([13*]**,[17])
In any pseudo-BCI algebra A the following are equivalent, for all x,y∈A:
(a) A is p-semisimple;
(b) if x≤y, then x=y;
(c) (x→y)⇝y=(x⇝y)→y=x;
(d) (x→1)⇝1=(x⇝1)→1=x;
(e) (x→1)⇝y=(y⇝1)→x;
(f) x→a=y→a implies x=y;
(g) x⇝a=y⇝a implies x=y;
(h) A=At(A);
(i) (A,⋅,−1,1) is a group, where x⋅y=(x→1)⇝y=(y⇝1)→x, x−1=x→1=x⇝1,
x→y=y⋅x−1 and x⇝y=x−1⋅y, for all x,y∈A.*
Proposition 2.4**.**
([14])*
In any pseudo-BCI algebra A the following are equivalent, for all a,x,y∈A:
(a) a∈At(A);
(b) (a→x)⇝x=(a⇝x)→x=a;
(c) x→a=(a→x)⇝1;
(d) x⇝a=(a⇝x)→1;
(e) x→a=(a→y)⇝(x→y);
(f) x⇝a=(a⇝y)→(x⇝y);
(g) x→a=((x→a)⇝y)→y;
(h) x⇝a=((x⇝a)→y)⇝y;
(i) x→a=(a→1)⇝(x→1);
(j) x⇝a=(a⇝1)→(x⇝1);
(k) (a→1)⇝1=(a⇝1)→1=a.*
Corollary 2.5**.**
If a∈At(A), then x→a,x⇝a∈At(A), for all x∈A.
Proof.
It follows taking y:=1 in Proposition 2.4(g),(h) and applying (k).
∎
Let (A,→,⇝,1) be a pseudo-BCI algebra. Denote: x⋓1y=(x→y)⇝y and x⋓2y=(x⇝y)→y,
for all x,y∈A.
Lemma 2.6**.**
*In any pseudo-BCI algebra A the following hold for all x,x1,x2,y∈A:
(1) 1⋓1x=1⋓2x=1;
(2) x≤y iff x⋓1y=y iff x⋓2y=y;
(3) x⋓1x=x⋓2x=x;
(4) if A is p-semisimple, then x⋓11=x⋓21=x;
(5) x≤x⋓1y and x≤x⋓2y;
(6) x1≤x2 implies x1⋓1y≤x2⋓1y and x1⋓2y≤x2⋓2y;
(7) x⋓1y→y=x→y and x⋓2y⇝y=x⇝y.*
Proof.
The proof is straightforward.
∎
A pseudo-BCK algebra A is said to be commutative if it satisfies the following identities
for all x,y∈A:
(comm1) x⋓1y=y⋓1x,
(comm2) x⋓2y=y⋓2x.
Note that if a pseudo-BCI algebra A satisfies (comm1) and (comm2) for all x,y∈A, then A is a
pseudo-BCK algebra. Indeed, x≤(x→1)⇝1=(1→x)⇝x=1, for all x∈A.
A pseudo-BCI algebra satisfying (comm1) and (comm2) for all x and y belonging to the same branch is
called branchwise commutative, and it is called commutative if it satisfies the quasi-identities
(comm3) x⋓1y=x⋓2y=x, whenever y≤x.
It was proved in [12] that a pseudo-BCI algebra is commutative if and only if it is branchwise commutative.
Moreover, each branch of a commutative pseudo-BCI algebra is a semilattice with the join ∨ defined by
x∨y=x⋓1y=x⋓2y.
It is known that any p-semisimple pseudo-BCI algebra is commutative ([12]).
A subset D of a pseudo-BCI algebra A is called a deductive system of A if it satisfies
the following axioms:
(ds1) 1∈D,
(ds2) x∈D and x→y∈D imply y∈D.
A subset D of A is a deductive system if and only if it satisfies (ds1) and the axiom:
(ds2′) x∈D and x⇝y∈D imply y∈D.
Denote by DS(A) the set of all deductive systems of A.
A deductive system D of A is proper if D=A.
A deductive system D of a pseudo-BCK algebra A is said to be compatible if it satisfies the condition:
(ds3) for all x,y∈A, x→y∈D iff x⇝y∈D.
Denote by DSc(A) the set of all compatible deductive systems of A.
A deductive system is closed if it is subalgebra of A.
Denote by CON(A) the set of all congruences of A.
We say that θ∈CON(A) is a relative congruence of A if the quotient algebra
(A/θ,→,⇝,[1]θ) is a pseudo-BCI algebra.
It was proved in [16] that the relative congruences of A correspond one-to-one to closed compatible deductive
systems of A.
For details regarding the deductive systems and congruence relations on a pseudo-BCI algebra we refer the reader to [16].
3. Implicative derivation operators on pseudo-BCI algebras
In this section we define the type I and type II implicative derivations on pseudo-BCI algebras, and we
investigate their properties. The notions of regular, isotone and idempotent implicative derivations are introduced,
and a characterization of regular implicative derivations of type II is given.
It is also proved that any regular implicative derivation of type II is isotone.
If d is a regular implicative derivation of type II on a pseudo-BCI algebra A such that its kernel coincides
with the pseudo-BCK part of A, then it is proved that d is idempotent.
Finally, given an implicative derivation d on A, we define the notion of a d-invariant deductive system of A,
and we prove that d is a regular implicative derivation of type II if and only if every deductive system of A
is d-invariant.
Definition 3.1**.**
Let (A,→,⇝,1) be a pseudo-BCI algebra. A mapping d:A⟶A is called an
implicative derivation operator of type I or a type I implicative derivation operator or a
type I implicative derivation on A if it satisfies the following conditions for all x,y∈A:
(idop1) d(x→y)=(x→d(y))⋓2(d(x)→y).
(idop2) d(x⇝y)=(x⇝d(y))⋓1(d(x)⇝y).
Definition 3.2**.**
Let (A,→,⇝,1) be a pseudo-BCI algebra. A mapping d:A⟶A is called an
implicative derivation operator of type II or a type II implicative derivation operator or a
type II implicative derivation on A if it satisfies the following conditions for all x,y∈A:
(idop3) d(x→y)=(d(x)→y)⋓2(x→d(y)),
(idop4) d(x⇝y)=(d(x)⇝y)⋓1(x⇝d(y)).
Let A be a pseudo-BCI algebra. Denote:
IDOP(I)(A) the set of all implicative derivation operators of type I on A,
IDOP(II)(A) the set of all implicative derivation operators of type II on A,
IDOP(A)=IDOP(I)(A)∩IDOP(II)(A).
A map d∈IDOP(A) is called an implicative derivation on A.
In what follows we will denote dx instead of d(x).
Example 3.3**.**
Let A be a pseudo-BCI algebra.
If IdA:A⟶A, defined by IdA(x)=x for all x∈A, then IdA∈IDOP(A).
Example 3.4**.**
Consider the structure (A,→,⇝,1), where the operations → and ⇝ on A={a,b,c,d,1}
are defined as follows:
[TABLE]
Then (A,→,⇝,1) is a pseudo-BCI algebra ([18]).
Consider the maps d1,d2,d3:A⟶A given in the table below:
[TABLE]
One can check that IDOP(I)(A)=IDOP(II)(A)={d1,d2,d3}.
Proposition 3.5**.**
*Let A be a pseudo-BCI algebra. Then the following hold for all x∈A:
(1) if d∈IDOP(I)(A), then dx=dx⋓1x=dx⋓2x;
(2) if d∈IDOP(II)(A), then dx=x⋓1dx=x⋓2dx iff d1=1.*
Proof.
(1) Consider d∈IDOP(I)(A) and x∈A. Applying Lema 2.2(8),(7) we get:
dx=d(1→x)=(1→dx)⋓2(d1→x)=dx⋓2(d1→x)
=(dx⇝(d1→x))→(d1→x)
=(d1→(dx⇝x))→(d1→x)
≥(dx⇝x)→x=dx⋓2x.
On the other hand dx≤dx⋓2x, hence dx=dx⋓2x.
Similarly we have:
dx=d(1⇝x)=(1⇝dx)⋓1(d1⇝x)=dx⋓1(d1⇝x)
=(dx→(d1⇝x))⇝(d1⇝x)
=(d1⇝(dx→x))⇝(d1⇝x)
≥(dx→x)⇝x=dx⋓1x.
Since dx≤dx⋓1x, it follows that dx=dx⋓1x.
(2) Let d∈IDOP(II)(A) and let x∈A. Then:
dx=d(1→x)=(d1→x)⋓2(1→dx)=(d1→x)⋓2dx and
dx=d(1⇝x)=(d1⇝x)⋓1(1⇝dx)=(d1⇝x)⋓1dx.
If d1=1, then dx=x⋓1dx=x⋓2dx.
Conversely, if dx=x⋓1dx=x⋓2dx for all x∈A, then d1=1⋓1d1=1⋓2d1=1.
∎
For any x∈A, define φ:A⟶A by φ(x)=φx=x⋓11.
By Lemma 2.2(11) we have φx=x⋓11=x⋓21=(x→1)→1=(x⇝1)⇝1.
Lemma 3.6**.**
*In any pseudo-BCI algevra A the following hold for any x∈A:
(1) φx∈At(A);
(2) φx→x,φx⇝x∈K(A);
(3) x∈K(A) implies φx=1.*
Proof.
(1) Since by Lemma 2.6(7), (φx→1)⇝1=(x⋓11→1)⇝1=(x→1)⇝1=φx and
(φx⇝1)→1=(x⋓21⇝1)→1=(x⇝1)→1=φx, applying Proposition 2.4(b) it
follows that φx∈At(A).
(2) From x≤φx we have φx→1≤x→1, that is (φx→1)⇝(x→1)=1.
Applying Lemma 2.2(12) we get (φx→x)→1=1, hence φx→x≤1, that
is φx→x∈K(A). Similarly, φx⇝x∈K(A).
(3) It is obvious.
∎
Proposition 3.7**.**
Let A be a pseudo-BCI algebra and let dφ:A⟶A, defined by
dx=φx for any x∈A. Then dφ∈IDOP(I)(A).
Proof.
Let A be a pseudo-BCI algebra and let x,y∈A. By Lemma 3.6, φy∈At(A) and by
Corollary 2.5, x→φy,x⇝φy∈At(A).
Applying Lemma 2.2(12) and Proposition 2.4(b) we get:
d(x→y)=φx→y=((x→y)→1)⇝1=((x→1)⇝(y→1))⇝1
=((x→1)⇝1)→((y→1)⇝1)=((x→1)⇝1)→φy
=((x→1)⇝1)→((φy→1)⇝1)=((x→1)⇝(φy→1))⇝1
=((x→φy)→1)⇝1=x→φy=((x→φy)⇝(φx→y))→(φx→y)
=(x→φy)⋓2(φx→y)=(x→dy)⋓2(dx→y).
Similarly we have:
d(x⇝y)=φx⇝y=((x⇝y)→1)⇝1=((x⇝y)⇝1)→1
=((x⇝1)→(y⇝1))→1=((x⇝1)→1)⇝((y⇝1)→1)
=((x⇝1)→1)⇝φy=((x⇝1)→1)⇝((φy⇝1)→1)
=((x⇝1)→(φy⇝1))→1=((x⇝φy)⇝1)→1=x⇝φy
=((x⇝φy)→(φx⇝y))⇝(φx⇝y)=(x⇝φy)⋓1(φx⇝y)
=(x⇝dy)⋓1(dx⇝y).
We conclude that dφ∈IDOP(I)(A).
∎
Theorem 3.8**.**
Let A be a commutative pseudo-BCI algebra, and let dφ:A⟶A, defined by dφx=φx for any x∈A. Then dφ∈IDOP(A).
Proof.
By Proposition 3.7, we have dφ∈IDOP(I)(A), and let x,y∈A.
As we mentioned in the proof of Proposition 3.7,
φx,φy,x→φy,x⇝φy∈At(A). Then we have:
x→φy=((x→φy)→1)⇝1=((x→1)⇝(φy→1))⇝1
=((x→1)⇝1)→((φy→1)⇝1)=φx→φy,
hence x→φy∈V(φx→φy). Moreover:
φx→y≤((φx→y)→1)⇝1=((φx→1)⇝(y→1))⇝1
=((φx→1)⇝1)→((y→1)⇝1)=φx→φy,
that is φx→y∈V(φx→φy).
Since A is commutative, it follows that A is branch commutative, and in the similar way to the proof
of Proposition 3.7 we get:
(dφx→y)⋓2(x→dφy)=(φx→y)⋓2(x→φy)=(x→φy)⋓2(φx→y)
=φx→y=dφ(x→y).
Similarly we have:
(dφx⇝y)⋓1(x⇝dφy)=(φx⇝y)⋓1(x⇝φy)=(x⇝φy)⋓1(φx⇝y)
=φx⇝y=dφ(x⇝y),
hence dφ∈IDOP(II)(A). We conclude that dφ∈IDOP(A).
∎
Definition 3.9**.**
Let A be a pseudo-BCI algebra A. A type I or a type II derivation d on A is
said to be:
(1) regular if d1=1;
(2) isotone if x≤y implies dx≤dy;
(3) idempotent if d2=d, where d2=d∘d.
Example 3.10**.**
Consider the pseudo-BCI algebra A and its derivations from Example 3.4.
Then:
(1) dφ=d3∈IDOP(A).
(2) d1 and d3 are regular implicative derivations, while d2 is not regular.
(3) d1, d2, d3 are isotone.
(4) d1 and d3 are idempotent.
Remark 3.11**.**
In Theorem 3.8 the pseudo-BCI algebra A need not be commutative.
Indeed, the pseudo-BCI algebra from Example 3.4 is not commutative, but
dφ=d3∈IDOP(A).
Let A be a pseudo-BCI algebra. Denote:
RIDOP(I)(A) the set of all regular implicative derivation
operators of type I on A,
RIDOP(II)(A) the set of all regular implicative derivation
operators of type II on A,
RIDOP(A)=RIDOP(I)(A)∩RIDOP(II)(A).
Proposition 3.12**.**
*Let A be a pseudo-BCI algebra and let d∈RIDOP(II)(A).
Then the following hold for all x,y∈A:
(1) x≤dx;
(2) dx→y≤dx→dy≤x→dy=d(x→y) and dx⇝y≤dx⇝dy≤x⇝dy=d(x⇝y);
(3) d is isotone;
(4) Ker(d)={x∈A∣dx=1} is a subalgebra of A;
(5) Ker(d)⊆K(A);
(6) d(K(A))⊆K(A);
(7) φx and dx belong to the same branch of A;
(8) φx→dx,φx⇝dx∈K(A).*
Proof.
(1) By Lemma 2.6(5) and Proposition 3.5(2), x≤x⋓1dx=dx.
(2) From y≤dy we have dx→y≤dx→dy, and by x≤dx we get dx→dy≤x→dy.
It follows that:
dx→y≤dx→dy≤x→dy=1→(x→dy)
=((dx→y)⇝(x→dy))→(x→dy)
=(dx→y)⋓2(x→dy)=d(x→y).
Similarly, dx⇝y≤dx⇝dy≤x⇝dy=d(x⇝y).
(3) Let x,y∈A such that x≤y, that is x→y=1.
Then we have dx→y≤(dx→y)⋓2(x→dy)=d(x→y)=d1=1. It follows that dx≤y≤dy, hence d
is isotone.
(4) Since d1=1, it follows that 1∈Ker(d). Let x,y∈Ker(d), that is dx=dy=1.
Then we have 1=dx→dy≤d(x→y), hence d(x→y)=1, that is x→y∈Ker(d).
Similarly, x⇝y∈Ker(d), thus Ker(d) is a subalgebra of A.
(5) For any x∈Ker(d) we have x≤dx=1, that is x∈K(A). Hence Ker(d)⊆K(A).
(6) If x∈d(K(A)), there exists x′∈K(A) such that x=dx′.
Since x′≤1 and x′≤dx′=x, applying Lemma 2.2 we get
x=1→x≤x′→x=1, that is x∈K(A). It follows that d(K(A))⊆K(A).
(7) From x≤dx, it follows that x and dx are in a branch V1 of A, while from x≤φx we get that x and φx are in a branch V2 of A. Since x∈V1∩V2, it follows that V1 and V2
coincide, hence x, φx and dx belong to the same branch of A.
(8) According to (7), there exists a∈At(A) such that φx,dx∈V(a).
From φx≤a, dx≤a we get φx→dx≤φx→a=1, that is φx→dx∈K(A).
Similarly, φx⇝dx∈K(A).
∎
Proposition 3.13**.**
Let A be a pseudo-BCI algebra and let d1,d2∈RIDOP(II)(A)
such that d2 is idempotent and d1≤d2 (that is d1x≤d2x, for all x∈A). Then d2∘d1=d2.
Proof.
Let x∈A. By Proposition 3.12(1),(3), d2x≤d2d1x=(d2∘d1)(x), so
d2≤d2∘d1. Moreover, since d1x≤d2x we have d2d1x≤d2d2x=d22x=d2x, that is
d2∘d1≤d2. Hence d2∘d1=d2.
∎
Proposition 3.14**.**
Let A be a pseudo-BCI algebra and let d∈RIDOP(II)(A).
Then Ker(d)=K(A) if and only if d=dφ.
Proof.
Assume that Ker(d)=K(A) and let x∈A.
Since by Lemma 3.6, φx→x∈K(A), we have d(φx→x)=1.
Applying Proposition 3.12(2) we get 1=d(φx→x)=φx→dx, hence φx≤dx.
On the other hand φx∈At(A), thus dx=φx.
Conversely, assume that dx=φx for all x∈A. Since for any x∈K(A), dx=φx=1, we have
x∈Ker(d), so that K(A)⊆Ker(d).
By Proposition 3.12(5), Ker(d)⊆K(A), hence Ker(d)=K(A).
∎
Corollary 3.15**.**
Let A be a pseudo-BCI algebra and let d∈RIDOP(II)(A) such that
Ker(d)=K(A). Then d is idempotent.
Proof.
According to Proposition 3.14, for any x∈A we have:
d2x=ddx=dφdx=(dx→1)⇝1=(φx→1)⇝1=φx=dx, hence d2=d.
∎
Corollary 3.16**.**
Let A be a pseudo-BCI algebra and let d1,d2∈RIDOP(II)(A)
such that d1≤d2 and Ker(d1)=K(A). Then d2∘d1=d2.
Proof.
Since d1≤d2, we have Ker(d1)⊆Ker(d2), that is Ker(d1)=Ker(d2)=K(A).
According to Corollary 3.15, d1 and d2 are idempotent, and applying Proposition 3.13
it follows that d2∘d1=d2.
∎
Example 3.17**.**
With the notations from Example 3.4, we have:
(1) K(A)={a,b,c,1};
(2) Ker(d3)={a,b,c,1}=K(A) and d32=d3;
(3) d1(K(A))=K(A) and d3(K(A))={1}⊆K(A).
Proposition 3.18**.**
*Let A be a pseudo-BCI algebra and let d∈IDOP(A).
If there exists a∈A such that a≤dx for all x∈A, then:
(1) d∈RIDOP(A);
(2) A is a pseudo-BCK algebra.*
Proof.
(1) From a≤dx, we have a→dx=a⇝dx=1 for all x∈A, so a→d(a→x)=1 and a→d1=a⇝d1=1.
Then we get 1=a→d(a→x)=a→(a→dx)⋓2(da→x)=a→(1⋓2(da→x))=a→1 and
similarly, a⇝1=1. It follows that
d1=d(a→1)=(da→1)⋓1(a→d1)=(da→1)⋓11=((da→1)→1)⇝1=1 (from 1∈At(A) we have
da→1∈At(A), hence (da→1)→1)⇝1=1). Hence d1=1, so d∈RIDOP(A).
(2) Since d is regular, we have x≤dx, thus dx→1≤x→1 for all x∈A. Then we have:
x→1≥dx→1=dx→(a⇝dx)=a⇝(dx→dx)=a⇝1=1, hence x→1=1.
We conclude that A is a pseudo-BCK algebra.
∎
Lemma 3.19**.**
*Let A be a pseudo-BCI algebra and let d∈IDOP(I)(A). Then:
(1) d1∈At(A);
(2) da=(a→1)→d1=(a→1)⇝d1=a⋅d1, for all a∈At(A);
(3) d(dx→x)=d(dx⇝x)=1, for all x∈A.*
Proof.
(1) By Proposition 3.5(1), d1=d1⋓11=d1⋓21=(d1→1)⇝1=(d1⇝1)→1, hence
by Proposition 2.4(k), d1∈At(A).
(2) Applying Lemma 2.2(8),(7),(2) we have:
da=d((a→1)⇝1)=((a→1)⇝d1)⋓1(d(a→1)⇝1)
=(((a→1)⇝d1)→(d(a→1)⇝1))⇝(d(a→1)⇝1)
=(d(a→1)⇝(((a→1)⇝d1)→1))⇝(d(a→1)⇝1)
≥(((a→1)⇝d1)→1)⇝1≥(a→1)⇝d1.
Since by (1), d1∈At(A), according to Corollary 2.5, (a→1)⇝d1∈At(A).
It follows that da=(a→1)⇝d1. Similarly, da=(a→1)→d1. Obviously da=a⋅d1.
(3) For any x∈A we have d(dx→x)=(dx→dx)⋓2(x→ddx)=1⋓2(x→ddx)=1 and
similarly, d(dx⇝x)=1.
∎
Lemma 3.20**.**
*Let A be a pseudo-BCI algebra and let d∈IDOP(II)(A). Then:
(1) d1→x≤dx and d1⇝x≤dx, for all x∈A;
(2) da=d1→a=d1⇝a=d1⋅a, for all a∈At(A).*
Proof.
(1) For all x∈A we have dx=d(1→x)=(d1→x)⋓2dx≥d1→x and
dx=d(1⇝x)=(d1⇝x)⋓1dx≥d1⇝x.
(2) From d1→a≤da, since a∈At(A) we get d1→a∈At(A), hence da=d1→a.
Similarly, da=d1⇝a. Since d1=d1→1, we have da=d1⇝a=(d1→1)⇝a=d1⋅a.
∎
Proposition 3.21**.**
*Let A be a pseudo-BCI algebra and let
d∈IDOP(I)(A)∪IDOP(II)(A). Then:
(1) d(At(A))⊆At(A);
(2) if d1∈K(A), then d1=1;
(3) d(a⋅b)=da⋅(d1→1)⋅db, for all a,b∈At(A);
(4) d∣At(A)=IdAt(A) iff d1=1.*
Proof.
(1) Let a∈At(A) and let d∈IDOP(I)(A), by Lemma 3.19(1), d1∈At(A), so
(a→1)→d1∈At(A). Hence by Lemma 3.19(2), da∈At(A).
If d∈IDOP(II)(A), the assertion follows applying Lemma 3.20(2).
(2) Since d1≤1 and by (1), d1∈At(A), we get d1=1.
(3) Let a,b∈At(A), that is a⋅b=(a→1)⇝b∈At(A).
If d∈IDOP(I)(A), by Lemma 3.19(2) we have:
a⋅b=(a⋅b)⋅d1=(a⋅d1)⋅(d1)−1⋅(b⋅d1)=da⋅(d1→1)⋅db.
For the case of d∈IDOP(II)(A), applying Lemma 3.20(2) we get:
a⋅b=d1⋅(a⋅b)=(d1⋅a)⋅(d1)−1⋅(d1⋅b)=da⋅(d1→1)⋅db.
(4) If d∣At(A)=IdAt(A), then d1=1.
Conversely, if d∈IDOP(I)(A)∪IDOP(II)(A) such that d1=1, then for all x∈At(A)
we have dx=x⋅d1=x⋅1=x and dx=d1⋅x=1⋅x=x, respectively. Hence d∣At(A)=IdAt(A).
∎
Example 3.22**.**
Let A be the pseudo-BCI algebra and its derivations from Example 3.4.
We can see that At(A)={d,1}, RIDOP(II)(A)={d1,d3} and
d1∣At(A)=d3∣At(A)=IdAt(A).
Moreover, for each d∈{d1,d2,d3} we have d(At(A))={d,1}=At(A).
Theorem 3.23**.**
Let A be a pseudo-BCI algebra and let d:A⟶A.
Then d∈RIDOP(II)(A) if and only if d1=1 and d(x→y)=x→dy, d(x⇝y)=x⇝dy,
for all x,y∈A.
Proof.
Let d∈RIDOP(II)(A), and let x,y∈A.
Clearly d1=1, and according to Proposition 3.12(2), d(x→y)=x→dy and d(x⇝y)=x⇝dy.
Conversely, let d:A⟶A such that d1=1 and d(x→y)=x→dy, d(x⇝y)=x⇝dy,
for all x,y∈A. Taking x:=d1, y:=1 we get d1=d(d1→1)=d1→d1=1.
For y:=x, we have x→dx=d(x→x)=d1=1, so x≤dx for all x∈A.
From x≤dx and y≤dy we get dx→y≤x→y≤x→dy and dx⇝y≤x⇝y≤x⇝dy.
It follows that d(x→y)=x→dy=1→(x→dy)=((dx→y)⇝(x→dy))→(x→dy)=(dx→y)⋓2(x→dy).
Similarly, d(x⇝y)=(dx⇝y)⋓1(x⇝dy), hence d∈RIDOP(II)(A).
∎
Proposition 3.24**.**
Let A be a pseudo-BCI algebra and let d∈IDOP(I)(A) such that
Im(d)⊆At(A). Then d(x→y)=x→dy and d(x⇝y)=x⇝dy, for all x,y∈A.
Proof.
Let x,y∈A. Since dy∈At(A), according to Corollary 2.5, x→dy∈At(A).
From x→dy≤(x→dy)⋓2(dx→y)=d(x→y), it follows that d(x→y)=x→dy.
Similarly, d(x⇝y)=x⇝dy.
∎
Remark 3.25**.**
If d∈RIDOP(I)(A)∪RIDOP(II)(A) such that
d(x→y)=dx→y or d(x⇝y)=dx⇝y for all x,y∈A, then d=IdA.
Indeed, dx=d(1→x)=d1→x=1→x=x for all x∈A, that is d=IdA. Similarly for d(x⇝y)=dx⇝y.
Definition 3.26**.**
Let A be a pseudo-BCI algebra and let d∈IDOP(A).
Then D∈DS(A) is said to be d-invariant if d(D)⊆D.
Proposition 3.27**.**
Let A be a pseudo-BCI algebra and let
d∈IDOP(I)(A)∪IDOP(II)(A). If every deductive system of A is d-invariant,
then d is regular.
Proof.
Let d∈IDOP(I)(A)∪IDOP(II)(A) and assume that every
D∈DS(A) is d-invariant. Since {1}∈DS(A), it follows that
d({1})⊆{1}, hence d(1)=1, that is d is
regular.
∎
Proposition 3.28**.**
Let A be a pseudo-BCI algebra and let d∈RIDOP(II)(A).
Then every deductive system of A is d-invariant.
Proof.
Assume that d∈RIDOP(II)(A). Let D∈DS(A) and let y∈d(D), that is there exists
x∈D such that y=dx. Since by Proposition 3.12, x→y=x→dx=1∈D, it follows that y∈D.
Hence d(D)⊆D, that is D is d-invariant.
∎
Corollary 3.29**.**
Let A be a pseudo-BCI algebra and let d∈IDOP(II)(A).
Then d∈RIDOP(II)(A) if and only if every deductive system of A is d-invariant.
Example 3.30**.**
Let A be the pseudo-BCI algebra and its derivations from Example 3.4.
We can see that DS(A)={{1},{c,1},{a,b,c,1},A} and consider d2∈IDOP(II)(A), but d2∈/RIDOP(II)(A). For D={c,1}∈DS(A), one can easily check that
d2(D)={d}⊈D, hence D is not d2-invariant.
On the other hand, for d1,d3⊆RIDOP(II)(A) we have d1(D)⊆D
and d3(D)⊆D, so that D is d1-invariant and d3-invariant.
Example 3.31**.**
Let A be a pseudo-BCI algebra and let d∈IDOP(II)(A).
According to [16, Prop. 12], K(A) is a closed compatible deductive system of A.
By Proposition 3.12, d(K(A))⊆K(A), hence K(A) is d-invariant.
4. Implicative derivation operators on p-semisimple pseudo-BCI algebras
In this section we investigate the particular case of implicative derivations on the p-semisimple
pseudo-BCI algebras, and we prove that a pseudo-BCI algebra A is p-semisimple if and only if the only regular
derivation of type II on A is the identity map. It is also proved that a pseudo-BCI algebra A is p-semisimple
if and only if the kernel of any regular implicative derivation d of type II on A is the set {1}.
As a corollary it is proved that, for any pseudo-BCI algebra A, the only regular implicative derivation of
type II on A/K(A) is the identity map.
It is also proved that the set of all implicative derivations of a p-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition.
Theorem 4.1**.**
*If A is a pseudo-BCI algebra, the following are equivalent:
(a) A is p-semisimple;
(b) Ker(d)={1} for all d∈RIDOP(II)(A);
(c) RIDOP(II)(A)={IdA}.*
Proof.
(a)⇒(b) Let A be a p-semisimple pseudo-BCI algebra, that is K(A)={1}, and let
d∈RIDOP(II)(A). According to Proposition 3.12(5), Ker(d)⊆K(A), hence Ker(d)={1}.
(b)⇒(a) Assume that Ker(d)={1} for all d∈RIDOP(II)(A).
Consider dφ:A⟶A, defined by dφx=φx for any x∈A.
Since any p-semisimple pseudo-BCI algebra is commutative, applying Theorem 3.8 it follows that
dφ∈IDOP(II)(A). Taking into consideration that dφ1=1, we have
dφ∈RIDOP(II)(A), so that Ker(dφ)={1}.
Moreover, for any x∈K(A), dφx=(x→1)⇝1=1, that is x∈Ker(dφ).
Thus K(A)⊆Ker(dφ), and applying Proposition 3.12(5) we get
K(A)=Ker(dφ)={1}, hence A is p-semisimple.
(b)⇒(c) If d∈RIDOP(II)(A), then by (b) we have Ker(d)={1}.
Since (b)⇒(a) and by Proposition 2.3, At(A)=A, applying Proposition 3.21
we get d=IdA.
(c)⇒(b) Let d∈RIDOP(II)(A), so by (c) we have d=IdA, that is Ker(d)={1}.
∎
Corollary 4.2**.**
If A is a pseudo-BCI algebra, then RIDOP(II)(At(A))={IdA}.
Proof.
According to [13, Th. 4.13], At(A) is a p-semisimple pseudo-BCI subalgebra of A, and by
Theorem 4.1 it follows that RIDOP(II)(At(A))={IdA}.
∎
Corollary 4.3**.**
If A is a pseudo-BCI algebra, then RIDOP(II)(A/K(A))={IdA}.
Proof.
According to [16, Prop.12, Th. 6], K(A) is a closed compatible deductive system of A and A/K(A) is a p-semisimple pseudo-BCI algebra. Applying Theorem 4.1, it follows that
RIDOP(II)(A/K(A))={IdA}.
∎
Example 4.4**.**
Consider the structure (A,→,⇝,1), where the operations → and ⇝ on A={a,b,c,d,e,1}
are defined as follows:
[TABLE]
Then (A,→,⇝,1) is a p-semisimple pseudo-BCI algebra ([16]), and one can check that:
(1) At(A)=A, K(A)={1} and A/K(A)=A;
(2) RIDOP(II)(A)=RIDOP(II)(At(A))=RIDOP(II)(A/K(A))={IdA}.
Example 4.5**.**
Let A be the pseudo-BCI algebra and its derivations from Example 3.4.
One can easily check that:
(1) At(A)={d,1};
(2) At(A) is a p-semisimple subalgebra of A;
(3) RIDOP(II)(At(A))=IdAt(A).
Remark 4.6**.**
The notion of a →medial pseudo-BCI algebra was defined in [31]
as a pseudo-BCI algebra A satisfying the identity (u⇝v)→(x⇝y)=(u⇝x)→(v⇝y), for all
x,y,u,v∈A. It was proved in [13] that a →medial pseudo-BCI algebra is a p-semisimple BCI-algebra,
so that by Theorem 4.1, RIDOP(II)(A)={IdA}.
Similarly for the case of a ⇝medial pseudo-BCI algebra which is a pseudo-BCI algebra A satisfying
the identity (u→v)⇝(x→y)=(u→x)⇝(v→y), for all x,y,u,v∈A ([36]).
Proposition 4.7**.**
Let A be a p-semisimple pseudo-BCI algebra and let
d1,d2∈IDOP(I)(A). Then d1∘d2∈IDOP(I)(A).
Proof.
According to Proposition 2.3(c), x⋓1y=x⋓2y=x for all x,y∈A.
If d1,d2∈IDOP(I)(A) we have:
(d1∘d2)(x→y)=d1d2(x→y)=d1((x→d2y)⋓2(d2x→y))
=d1(x→d2y)=(x→d1d2y)⋓2(d1x→d2y)
=x→d1d2y=(x→d1d2y)⋓2(d1d2x→y)
=(x→(d1∘d2)(y))⋓2((d1∘d2)(x)→y).
Similarly, (d1∘d2)(x⇝y)=(x⇝(d1∘d2)(y))⋓1((d1∘d2)(x)⇝y),
hence d1∘d2∈IDOP(I)(A).
∎
Proposition 4.8**.**
Let A be a p-semisimple pseudo-BCI algebra and let
d1,d2∈IDOP(II)(A). Then d1∘d2∈IDOP(II)(A).
Proof.
Let d1,d2∈IDOP(II)(A). Similar to Proposition 4.7 we get:
(d1∘d2)(x→y)=d1d2(x→y)=d1((d2x→y)⋓2(x→d2y))
=d1(d2x→y)=(d1d2x→y)⋓2(d2x→d1y)
=d1d2x→y=(d1d2x→y)⋓2(x→d1d2y)
=((d1∘d2)(x)→y)⋓2(x→(d1∘d2)(y)).
Similarly, (d1∘d2)(x⇝y)=((d1∘d2)(x)⇝y)⋓1(x⇝(d1∘d2)(y)),
thus d1∘d2∈IDOP(II)(A).
∎
Proposition 4.9**.**
Let A be a p-semisimple pseudo-BCI algebra and let
d1,d2∈IDOP(A). Then d1∘d2=d2∘d1.
Proof.
Let d1,d2∈IDOP(A). Using the definitions of type I and type II derivations we get:
(d1∘d2)(x)=(d1∘d2)(1→x)=d1((1→d2x)⋓2(d21→x))
=d1(1→d2x)=(d11→d2x)⋓2(1→d1d2x)=d11→d2x.
(d2∘d1)(x)=(d2∘d1)(1→x)=d2((d11→x)⋓2(1→d1x))
=d2(d11→x)=(d11→d2x)⋓2(d2d11→x)=d11→d2x.
Hence (d1∘d2)(x)=(d2∘d1)(x) for all x∈A. It follows that d1∘d2=d2∘d1.
∎
Theorem 4.10**.**
If A is a p-semisimple pseudo-BCI algebra then
(IDOP(A),∘,IdA) is a commutative monoid.
Proof.
It follows by Propositions 4.7, 4.8, 4.9 taking into consideration that the
composition of functions is always associative.
∎
Remark 4.11**.**
In Theorem 4.10 the pseudo-BCI algebra A need not be p-semisimple.
Indeed, consider the pseudo-BCI algebra A and its derivations from Example 3.4.
Since a≤b and a=b, the condition (b) from Proposition 2.3 is not satisfied, so A
is not p-semisimple. On the other hand we have d1∘d2=d2∘d1=d2, d1∘d3=d3∘d1=d3,
d2∘d3=d3∘d2=d2, hence (IDOP(A),∘,IdA) is a commutative monoid.
Proposition 4.12**.**
Let A be a p-semisimple pseudo-BCI algebra and let
d1,d2∈IDOP(A). Define (d1→d2)(x)=d1x→d2x and (d1⇝d2)(x)=d1x⇝d2x,
for all x∈A. Then d1→d2=d2→d1 and d1⇝d2=d2⇝d1.
Proof.
Let d1,d2∈IDOP(A). We use again the fact that in any p-semisimple pseudo-BCI algebra A
we have x⋓1y=x⋓2y=x for all x,y∈A. Applying the definitions of type I and type II
derivations we get:
(d1∘d2)(1)=(d1∘d2)(x→x)=d1((x→d2x)⋓2(d2x→x))
=d1(x→d2x)=(d1x→d2x)⋓2(x→d1d2x)=d1x→d2x.
(d1∘d2)(x)=(d1∘d2)(x→x)=d1((d2x→x)⋓2(x→d2x))
=d1(d2x→x)=(d2x→d1x)⋓2(d1d2x→x)=d2x→d1x.
Hence (d1→d2)(x)=(d2→d1)(x)=(d1∘d2)(x) for all x∈A.
It follows that d1→d2=d2→d1. Similarly, d1⇝d2=d2⇝d1.
∎
5. Symmetric derivation operators on pseudo-BCI algebras
In this section we give a generalization of the concept of left derivation introduced in [2] for BCI-algebras. Two types of symmetric derivations on pseudo-BCI algebras are defined and studied, and the
relationship between implicative and symmetric derivations is investigated.
We prove that for the case of a p-semisimple pseudo-BCI algebra the sets of type II implicative derivations
and type II symmetric derivations are equal, while for a p-semisimple BCI-algebra this result is also valid for
the sets of type I implicative derivations and type I symmetric derivations. Finally, we show that a type I or type II symmetric derivation d on a pseudo-BCI algebra A is regular if and only if every deductive system of A is d-invariant.
Definition 5.1**.**
Let (A,→,⇝,1) be a pseudo-BCI algebra. A mapping d:A⟶A is called a
symmetric derivation operator of type I or a type I symmetric derivation operator or a
type I symmetric derivation on A if it satisfies the following conditions for all x,y∈A:
(sdop1) d(x→y)=(x→d(y))⋓2(y→d(x)).
(sdop2) d(x⇝y)=(x⇝d(y))⋓1(y⇝d(x)).
Definition 5.2**.**
Let (A,→,⇝,1) be a pseudo-BCI algebra. A mapping d:A⟶A is
called a symmetric derivation operator of type II or a type II symmetric derivation operator
or a type II symmetric derivation on A if it satisfies the following conditions for all x,y∈A:
(sdop3) d(x→y)=(d(x)→y)⋓2(d(y)→x).
(sdop4) d(x⇝y)=(d(x)⇝y)⋓1(d(y)⇝x).
Let A be a pseudo-BCI algebra. Denote:
SDOP(I)(A) the set of all symmetric derivation operators of type I on A,
SDOP(II)(A) the set of all symmetric derivation operators of type II on A,
SDOP(A)=SDOP(I)(A)∩SDOP(II)(A).
A map d∈SDOP(A) is called a symmetric derivation on A.
In what follows we will denote dx instead of d(x).
Similar to the case of implicative derivations we can define the regular symmetric derivations on pseudo-BCI algebras.
Denote by RSDOP(I)(A), RSDOP(II)(A) and RSDOP(A)
the set of all regular symmetric derivations from SDOP(I)(A), SDOP(II)(A) and
SDOP(A), respectively.
Proposition 5.3**.**
*Let (A,→,⇝,1) be a pseudo-BCI algebra and let d∈SDOP(I)(A).
The following hold for all x,y∈A:
(1) d1=x→dx=x⇝dx;
(2) dx=dx⋓1d1=dx⋓2d1;
If d is regular, then:
(3) x≤dx;
(4) dx∈At(A);
(5) dx=dx⋓1y=dx⋓2y.*
Proof.
(1) We have d1=d(x→x)=(x→dx)⋓2(x→dx)=x→dx and similarly, d1=x⇝dx.
(2) Using Lemmas 2.2(7),(8) and 2.6(5) we get:
dx=d(1⇝x)=dx⋓1(x⇝d1)=(dx→(x⇝d1))⇝(x⇝d1)
=(x⇝(dx→d1))⇝(x⇝d1)
≥(dx→d1)⇝d1=dx⋓1d1≥dx.
Hence dx=dx⋓1d1 and similarly, dx=dx⋓2d1.
(3) It follows from (1), since d1=1.
(4) Since d1=1, applying (2) we get dx=(dx→1)⇝1=(dx⇝1)→1, hence by
Proposition 2.4(k), dx∈At(A).
(5) It follows by Proposition 2.4(b), since dx∈At(A).
∎
Proposition 5.4**.**
*Let (A,→,⇝,1) be a pseudo-BCI algebra and let d∈SDOP(II)(A).
The following hold for all x,y∈A:
(1) d1=dx→x=dx⇝x;
(2) dx=φdx⋓1x=φdx⋓2x;
(3) dx=dx⋓1x=dx⋓2x;
(4) dx∈At(A);
(5) if d is regular, then d=IdA.*
Proof.
(1) We have d1=d(x→x)=(dx→x)⋓2(dx→x)=dx→x and similarly, dx=dx⇝x.
(2) Applying Lemma 2.2(7),(8) and (1) we get:
dx=d(1⇝x)=(d1⇝x)⋓1(dx⇝1)=((d1⇝x)→(dx⇝1))⇝(dx⇝1)
=(dx⇝((d1⇝x)→1))⇝(dx⇝1)
≥((d1⇝x)→1)⇝1=(((dx→x)⇝x)→1)⇝1
=(dx⋓1x→1)⇝1=φdx⋓1x.
On the other hand, by Lemmas 2.2(2) and 2.6(5) we have:
φdx⋓1x=(dx⋓1x→1)⇝1≥dx⋓1x≥dx.
We conclude that dx=φdx⋓1x. Similarly, dx=φdx⋓2x.
(3) By (2), dx=φdx⋓1x=(dx⋓1x)→1)⇝1≥dx⋓1x≥dx,
so that dx=dx⋓1x.
Similarly, dx=dx⋓2x.
(4) It follows by (2), since φx∈At(A), for all x∈A;
(5) Since d1=1, by (3) and (1) we get: dx=dx⋓1x=(dx→x)⇝x=d1⇝x=1⇝x=x, for all x∈A.
Hence d=IdA.
∎
Corollary 5.5**.**
*If d∈SDOP(II)(A), then the following hold for all x,y∈A:
(1) Im(d)⊆At(A);
(2) dx=dx⋓1y=dx⋓2y;
(3) x→dy,x⇝dy∈At(A).*
Proposition 5.6**.**
Let A be a pseudo-BCI algebra and let
d∈SDOP(I)(A)∪SDOP(II)(A). Then d(x⋅y)=dx⋅y=x⋅dy, for all
x,y∈At(A).
Proof.
Let d∈SDOP(I)(A) and let x,y∈At(A).
Since 1,x,y∈At(A), by Corollary 2.5 and Proposition 5.4,
x→1,y⇝1,dx,dy,d(x→1),d(y⇝1),d1,d1→x,d1⇝x∈At(A).
From x⋅y=(x→1)⇝y=(y⇝1)→x, using Proposition 2.4(b) we get:
d(x⋅y)=d((x→1)⇝y)=((x→1)⇝dy)⋓1(y⇝d(x→1))
=(x→1)⇝dy=x⋅dy
d(x⋅y)=d((y⇝1)→x)=((y⇝1)→dx)⋓2(x→d(y⇝1))
=(y⇝1)→dx=dx⋅y.
Similarly, for d∈SDOP(II)(A) we have:
d(x⋅y)=d((x→1)⇝y)=(d(x→1)⇝y)⋓1(dy⇝(x→1))
=d(x→1)⇝y=((dx→1)⋓2(d1→x))⇝y
=(dx→1)⇝y=dx⋅y.
d(x⋅y)=d((y⇝1)→x)=(d(y⇝1)→x)⋓2(dx→(y⇝1))
=d(y⇝1)→x=((dy⇝1)⋓1(d1⇝y))→x
=(dy⇝1)→x=x⋅dy.
Hence, in both cases d(x⋅y)=dx⋅y=x⋅dy.
∎
Proposition 5.7**.**
Let A be a pseudo-BCI algebra and let dφ:A⟶A, defined by
dx=φx for any x∈A. Then dφ∈SDOP(I)(A).
Proof.
Similar to the proof of Proposition 3.7, since x→φy∈At(A), we get:
d(x→y)=φx→y=x→φy=((x→φy)⇝(y→φx))→(y→φx)
=(x→φy)⋓2(y→φx)=(x→dy)⋓2(y→dx).
Similarly we have:
d(x⇝y)=φx⇝y=x⇝φy=((x⇝φy)→(y⇝φx))⇝(y⇝φx)
=(x⇝φy)⋓1(y⇝φx)=(x⇝dy)⋓1(y⇝dx).
We conclude that dφ∈SDOP(I)(A).
∎
Corollary 5.8**.**
IDOP(I)(A)∩SDOP(I)(A)=∅.
Proof.
If dφ:A⟶A, defined by dx=φx, for all x∈A, then by Propositions 3.7
and 5.7, dφ∈IDOP(I)(A)∩SDOP(I)(A).
∎
Example 5.9**.**
Consider the pseudo-BCI algebra (A,→,⇝,1) and the maps
d1,d2,d3:A⟶A from Example 3.4.
One can check that dφ=d3, SDOP(I)(A)={d2,d3} and SDOP(II)(A)={d3}.
Example 5.10**.**
Consider the structure (A,→,⇝,1), where the operations → and ⇝ on A={a,b,x,y,g,1}
are defined as follows:
[TABLE]
Then (A,→,⇝,1) is a pseudo-BCI algebra ([20]).
Consider the maps d1,d2,d3:A⟶A given in the table below:
[TABLE]
One can see that SDOP(I)(A)={d2,d3} and SDOP(II)(A)=∅.
We also mention that dφ=d3 and IDOP(I)(A)=IDOP(II)(A)={d1,d2,d3}.
Proposition 5.11**.**
*Let A be a p-semisimple pseudo-BCI algebra and let d∈SDOP(II)(A).
Then the following hold for all x,y∈A:
(1) d(x→y)=dx→y and d(x⇝y)=dx⇝y;
(2) x→dx=y→dy and x⇝dx=y⇝dy;
(3) x→dx=dy→y and x⇝dx=dy⇝y.*
Proof.
(1) It follows by Proposition 2.3(c) since A is p-semisimple.
(2) By (psBCI1) we have:
(x→dx)⇝((dx→y)⇝(x→y))=(y→dy)⇝((dy→x)⇝(y→x))(=1).
It follows by (1) that:
(x→dx)⇝(d(x→y)⇝(x→y))=(y→dy)⇝(d(y→x)⇝(y→x)).
Applying Proposition 5.4(1) we have:
d(x→y)⇝(x→y)=d(y→x)⇝(y→x)(=d1).
Hence (x→dx)⇝(d(x⇝y)→(x→y))=(y→dy)⇝(d(x→y)⇝(x→y)).
Finally applying Proposition 2.3(f) we get x→dx=y→dy.
Similarly, x⇝dx=y⇝dy.
(3) Since by Proposition 5.4(1), d1=dy→y, applying (2) we get
x→dx=1→d1=d1=dy→y, that is x→dx=dy→y. Similarly, x⇝dx=dy⇝y.
∎
Proposition 5.12**.**
If A is a p-semisimple pseudo-BCI algebra, then
SDOP(II)(A)=IDOP(II)(A).
Proof.
Let d∈SDOP(II)(A). By Proposition 5.11(1), since A is p-semisimple we have
d(x→y)=dx→y=(dx→y)⋓2(x→dy), that is d∈IDOP(II)(A).
Hence SDOP(II)(A)⊆IDOP(II)(A).
Conversely, if d∈IDOP(II)(A), then
d(x→y)=(dx→y)⋓2(x→dy)=dx→y=(dx→y)⋓2(dy→x), that is d∈SDOP(II)(A).
Thus IDOP(II)(A)⊆SDOP(II)(A).
We conclude that SDOP(II)(A)=IDOP(II)(A).
∎
Proposition 5.13**.**
If A is a p-semisimple BCI-algebra, then SDOP(I)(A)=IDOP(I)(A).
Proof.
Similar to [2, Th. 3.13], based on the fact that any p-semisimple BCI-algebra is medial.
∎
Example 5.14**.**
Consider the structure (A,→,1) and the maps d1,d2,d3:A⟶A, defined in the tables below:
[TABLE]
Then (A,→,1) is a p-semisimple BCI-algebra ([5]).
One can check that SDOP(I)(A)=IDOP(I)(A)={d1,d2,d3} and
SDOP(II)(A)=IDOP(II)(A)={d1}.
Theorem 5.15**.**
Let A be a pseudo-BCI algebra and let
d∈SDOP(I)(A)∪SDOP(II)(A). Then d is regular if and only
if every deductive system of A is d-invariant.
Proof.
Let d∈SDOP(I)(A)∪SDOP(II)(A) and assume that every
D∈DS(A) is d-invariant. Since {1}∈DS(A), it follows that
d({1})⊆{1}, hence d(1)=1, that is d is regular.
Conversely, let D∈DS(A) and let y∈d(D), that is there exists x∈D such that y=dx.
If d∈SDOP(I)(A), then by Proposition 5.3(3) we have x→y=x→dx=1∈D,
hence y∈D.
For d∈SDOP(II)(A), by Proposition 5.4(5) we get x→y=x→dx=x→x=1,
that is y∈D.
It follows that d(D)⊆D, hence D is d-invariant.
∎
6. Conclusions and future work
In this paper we introduce and study two concepts of implicative derivation operators on pseudo-BCI algebras:
type I implicative derivation defined by conditions (idop1), (idop2) and type II implicative derivation
defined by conditions (idop3), (idop4).
These conditions were required by the proof of Proposition 3.5, which is crucial for the results of
this paper. For the particular case of the pseudo-BCK algebras the above mentioned result is also valid for
another two types of implicative derivations:
− type III implicative derivation defined by the following conditions, for all x,y∈A:
(idop5) d(x→y)=(x→dy)⋓1(dx→y)
(idop6) d(x⇝y)=(x⇝dy)⋓2(dx⇝y),
− type IV implicative derivation defined by the following conditions, for all x,y∈A:
(idop7) d(x→y)=(dx→y)⋓1(x→dy)
(idop8) d(x⇝y)=(dx⇝y)⋓2(x⇝dy).
In what follows we give an example of these derivations, but the investigation of type III and type IV implicative derivations on pseudo-BCK algebras is the topic of another work ([10]).
Consider the structure (A,→,⇝,1), where the operations → and ⇝ on A={0,a,b,c,1}
are defined as follows:
[TABLE]
Then (A,→,⇝,1) is a pseudo-BCK algebra ([9]).
Consider the maps di:A⟶A, i=1,⋯,6, given in the table below:
[TABLE]
One can check that IDOP(I)(A)=IDOP(II)(A)=IDOP(IV)(A)={d1,d2,d5,d6} and
IDOP(III)(A)={d1,d2,d3,d4,d5,d6}.
We also introduce two types of symmetric derivations on pseudo-BCI algebras and study the relationship between
implicative and symmetric derivations.
The results presented in this paper could be extended to other “pseudo” algebras such as pseudo-BCH algebras, pseudo-BE algebras, pseudo-CI algebras, non-commutative residuated lattices.
As a generalization of the concept of derivations, the f-derivation has been defined for lattices ([6]), BCI-algebras ([24], [39], [48]), BE-algebras ([33]).
If (A,∗,0) is a BCI-algebra and f is an endomorphism of A, then a map d:A⟶A is called a left-right f-derivation if d(x∗y)=(dx∗f(y))∧(f(x)∗dy) and a right-left f-derivation
if d(x∗y)=(f(x)∗dy)∧(dx∗f(y)) for all x,y∈A, where x∧y=y∗(y∗x).
As another direction of research one could define and study the concept of f-derivations on pseudo-BCI algebras
and other “pseudo” algebras.
The concept of generalized derivations on BCI-algebras defined and investigated in [40], [38], [37] could be also introduced and studied for the case of pseudo-BCI algebras.
The BCI-algebras with product (BCI(P)-algebras) (A,⊙,→,1) were originally introduced by
Iseˊki ([29]) as BCI-algebras with condition (S).
The concept of multiplicative derivations on BCI(P)-algebras could be defined and studied as another
topic of future research.