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The notion of normal pseudo-BCI-algebras is studied and some characterizations of it are given. Extensions of pseudo-BCI-algebras are also considered. 1. Introduction. Among many algebraic structures, algebras of logic form important class of algebras. Examples of these are (pseudo-)MV-algebras, (pseudo-)BL-algebras, (pseudo-)BCK-algebras, (pseudo-)BCI-algebras and others. They are strongly connected with logic. For example, BCI-algebras introduced in [8] have connections with BCI-logic being the BCI-system in combinatory logic which has application in the language of functional programming. The notion of pseudo-BCI-algebras has been introduced in [1] as an extension of BCI-algebras. Pseudo-BCI-algebras are algebraic models of some extension of a non-commutative version of the BCI-logic (see [5] for details). These algebras have also connections with other algebras of logic such as pseudo-BCK-algebras, pseudo-BL-algebras and pseudo-MV-algebras. More about those algebras the reader can find in [7]. The paper is devoted to pseudo-BCI-algebras. In Section 2 we give the necessary material needed in the sequel and also some new results concerning p-semisimple part and branches of pseudo-BCI-algebras. In Section 3 we consider normal pseudo-BCI-algebras, that is, pseudo-BCI-algebras X, which are the sum of their pseudo-BCK-part K(X) and p-semisimple part 2010 Mathematics Subject Classification. 03G25, 06F35. Key words and phrases. Pseudo-BCI-algebra, normal pseudo-BCI-algebra, extension of pseudo-BCI-algebra. M (X). We illustrate this notion by interesting examples and give some characterizations of it. In this section we also construct a new pseudoBCI-algebra being the sum of a pseudo-BCK-algebra and a p-semisimple pseudo-BCI-algebra (Theorem 3.4). Finally, in Section 4 we study extensions of pseudo-BCI-algebras. 2. Preliminaries. A pseudo-BCI-algebra is a structure (X; , , ;, 1), where is a binary relation on a set X, and ; are binary operations on X and 1 is an element of X such that for all x, y, z X, we have (a1) x y (y z) ; (x z), x ; y (y ; z) (x ; z), (a2) x (x y) ; y, x (x ; y) y, (a3) x x, (a4) if x y and y x, then x = y, (a5) x y iff x y = 1 iff x ; y = 1. It is obvious that any pseudo-BCI-algebra (X; , , ;, 1) can be regarded as a universal algebra (X; , ;, 1) of type (2, 2, 0). Note that every pseudo-BCI-algebra satisfying x y = x ; y for all x, y X is a BCIalgebra. Every pseudo-BCI-algebra satisfying x 1 for all x X is a pseudoBCK-algebra. A pseudo-BCI-algebra which is not a pseudo-BCK-algebra will be called proper. Throughout this paper we will often use X to denote a pseudo-BCIalgebra. Any pseudo-BCI-algebra X satisfies the following, for all x, y, z X, (b1) if 1 x, then x = 1, (b2) if x y, then y z x z and y ; z x ; z, (b3) if x y and y z, then x z, (b4) x (y ; z) = y ; (x z), (b5) x y z iff y x ; z, (b6) x y (z x) (z y), x ; y (z ; x) ; (z ; y), (b7) if x y, then z x z y and z ; x z ; y, (b8) 1 x = 1 ; x = x, (b9) ((x y) ; y) y = x y, ((x ; y) y) ; y = x ; y, (b10) x y (y x) ; 1, x ; y (y ; x) 1, (b11) (x y) 1 = (x 1) ; (y ; 1), (x ; y) ; 1 = (x ; 1) (y 1), (b12) x 1 = x ; 1. If (X; , , ;, 1) is a pseudo-BCI-algebra, then, by (a3), (a4), (b3) and (b1), (X; ) is a poset with 1 as a maximal element. Note that a pseudoBCI-algebra has also other maximal elements. Proposition 2.1 ([4]). The structure (X; , , ;, 1) is a pseudo-BCIalgebra if and only if the algebra (X; , ;, 1) of type (2, 2, 0) satisfies the following identities and quasi-identity: (i) (ii) (iii) (iv) (v) (x y) ; [(y z) ; (x z)] = 1, (x ; y) [(y ; z) (x ; z)] = 1, 1 x = x, 1 ; x = x, x y = 1 & y x = 1 x = y. Example 2.2 ([4]). Let X = {a, b, c, d, e, f, 1} and define binary operations and ; on X by the following tables: a b c d e f 1 a 1 c e b d a a b d 1 a e c b b c e a 1 d b c c d b e c 1 a d d e c d b a 1 e e f a b d c e 1 f 1 a b d c e 1 1 ; a b c d e f 1 a 1 d b e c a a b c 1 e a d b b c b e 1 d a c c d e a c 1 b d d e d c a b 1 e e f a b d c e 1 f 1 a b d c e 1 1 Then (X; , ;, 1) is a (proper) pseudo-BCI-algebra. Observe that it is not a pseudo-BCK-algebra because a 1. Example 2.3 ([9]). Let Y1 = (-, 0] and let be the usual order on Y1 . Define binary operations and ; on Y1 by xy= x;y= 0 2y if x y, y arctan(ln( x )) if y < x, 0 if x y, - tan( x ) 2y if y < x ye for all x, y Y1 . Then (Y1 ; , , ;, 0) is a pseudo-BCK-algebra, and hence it is a nonproper pseudo-BCI-algebra. Example 2.4 ([3]). Let Y2 = R2 and define binary operations and ; and a binary relation on Y2 by (x1 , y1 ) (x2 , y2 ) = (x2 - x1 , (y2 - y1 )e-x1 ), (x1 , y1 ) ; (x2 , y2 ) = (x2 - x1 , y2 - y1 ex2 -x1 ), (x1 , y1 ) (x2 , y2 ) (x1 , y1 ) (x2 , y2 ) = (0, 0) = (x1 , y1 ) ; (x2 , y2 ) for all (x1 , y1 ), (x2 , y2 ) Y2 . Then (Y2 ; , , ;, (0, 0)) is a proper pseudoBCI-algebra. Notice that Y2 is not a pseudo-BCK-algebra because there exists (x, y) = (1, 1) Y2 such that (x, y) (0, 0). Example 2.5 ([3]). Let Y be the direct product of pseudo-BCI-algebras Y1 and Y2 from Examples 2.3 and 2.4, respectively. Then Y is a proper pseudo-BCI-algebra, where Y = (-, 0] × R2 and binary operations and ; and binary relation are defined on Y by (x1 , y1 , z1 ) (x2 , y2 , z2 ) = = (0, y2 - y1 , (z2 - z1 )e-y1 ) if x1 x2 , ( 2x2 arctan(ln( x2 )), y2 - y1 , (z2 - z1 )e-y1 ) if x2 < x1 , x1 (x1 , y1 , z1 ) ; (x2 , y2 , z2 ) = = (0, y2 - y1 , z2 - z1 ey2 -y1 ) (x2 e - tan( 2x 1 ) 2 x if x1 x2 , ey2 -y1 ) if x2 < x1 , , y2 - y1 , z2 - z1 (x1 , y1 , z1 ) (x2 , y2 , z2 ) x1 x2 and y1 = y2 and z1 = z2 . Notice that Y is not a pseudo-BCK-algebra because there exists (x, y, z) = (0, 1, 1) Y such that (x, y, z) (0, 0, 0). For any pseudo-BCI-algebra (X; , ;, 1) the set K(X) = {x X : x 1} is a subalgebra of X (called pseudo-BCK-part of X). Then (K(X); , ;, 1) is a pseudo-BCK-algebra. Note that a pseudo-BCI-algebra X is a pseudoBCK-algebra if and only if X = K(X). It is easily seen that for the pseudo-BCI-algebras X, Y1 , Y2 and Y from Examples 2.2, 2.3, 2.4 and 2.5, respectively, we have K(X) = {f, 1}, K(Y1 ) = Y1 , K(Y2 ) = {(0, 0)} and K(Y ) = {(x, 0, 0) : x 0}. We will denote by M (X) the set of all maximal elements of X and call it the p-semisimple part of X. Obviously, 1 M (X). Notice that M (X) K(X) = {1}. Indeed, if a M (X) K(X), then a 1 and, by the fact that a is maximal, a = 1. Moreover, observe that 1 is the only maximal element of a pseudo-BCK-algebra. Therefore, for a pseudo-BCK-algebra X, M (X) = {1}. In [2] and [3] there is shown that M (X) = {x X : x = (x 1) 1} and it is a subalgebra of X. Observe that for the pseudo-BCI-algebras X, Y1 , Y2 and Y from Examples 2.2, 2.3, 2.4 and 2.5, respectively, we have M (X) = {a, b, c, d, e, 1}, M (Y1 ) = {0}, M (Y2 ) = Y2 and M (Y ) = {(0, y, z) : y, z R}. Proposition 2.6. Let X be a pseudo-BCI-algebra. Then M (X) = {x 1 : x X}. Proof. We know that M (X) = {x X : x = (x 1) 1}. Since, by (b9) and (b12), for any x X, x 1 = ((x 1) 1) 1, we get that x 1 M (X) for any x X. Hence, {x 1 : x X} M (X). Now, let a M (X). Then, a = (a 1) 1. Putting x = a 1 X we obtain that a = x 1 for some x X and also M (X) {x 1 : x X}. Therefore, M (X) = {x 1 : x X}. Let X be a pseudo-BCI-algebra. For any a X we define a subset V (a) of X as follows V (a) = {x X : x a}. Note that V (a) is non-empty, because a a gives a V (a). Notice also that V (a) V (b) for any a, b X such that a b. If a M (X), then the set V (a) is called a branch of X determined by element a. The following facts are proved in [3]: (1) branches determined by different elements are disjoint, (2) a pseudo-BCI-algebra is a set-theoretic union of branches, (3) comparable elements are in the same branch. The pseudo-BCI-algebra Y1 from Example 2.3 has only one branch (as the pseudo-BCK-algebra) and the pseudo-BCI-algebra X from Example 2.2 has six branches: V (a) = {a}, V (b) = {b}, V (c) = {c}, V (d) = {d}, V (e) = {e} and V (1) = {f, 1}. Every {(x, y)} is a branch of the pseudo-BCI-algebra Y2 from Example 2.4, where (x, y) Y2 . For the pseudo-BCI-algebra Y from Example 2.5 the sets V ((0, a1 , a2 )) = {(x, a1 , a2 ) Y : x 0}, where (0, a1 , a2 ) M (X), are branches of Y . Proposition 2.7 ([2]). Let X be a pseudo-BCI-algebra and let x X and a, b M (X). If x V (a), then x b = a b and x ; b = a ; b. Proposition 2.8 ([2]). Let X be a pseudo-BCI-algebra and let x, y X. The following are equivalent: (i) x and y belong to the same branch of X, (ii) x y K(X), (iii) x ; y K(X). Proposition 2.9 ([3]). Let X be a pseudo-BCI-algebra and let x, y X. If x and y belong to the same branch of X, then x 1 = x ; 1 = y 1 = y ; 1. We have the following proposition. Proposition 2.10. Let X be a pseudo-BCI-algebra and let x, y X. The following are equivalent: (i) x and y belong to the same branch of X, (ii) x y K(X), (iii) x ; y K(X), (iv) x 1 = x ; 1 = y 1 = y ; 1. Proof. Let x, y X. By Propositions 2.8 and 2.9 and (b12) it is sufficient to prove that if x 1 = y 1, then x y K(X), that is, (iv) (ii). Assume that x 1 = y 1. Then, by (b11) and (b12), we have (x y) 1 = (x 1) ; (y 1) = 1, which means that x y 1. Hence, x y K(X) and the proof is complete. We also have the following proposition. Proposition 2.11. Let X be a pseudo-BCI-algebra and let x, y X. The following are equivalent: (i) x and y belong to the same branch of X, (ii) x a = y a for all a M (X), (ii') x ; a = y ; a for all a M (X), (iii) x a y a for all a M (X), (iii') x ; a y ; a for all a M (X). Proof. (i) (ii): Assume that x, y V (b) for some b M (X). Then for any a M (X), by Proposition 2.7, we get x a = b a = y a, that is, (ii) holds. (ii) (i): If x a = y a for all a M (X), then putting a = 1 we get x 1 = y 1. Now, by Proposition 2.10, we obtain (i). (ii) (iii): Obvious. (iii) (ii): Let x a y a for all a M (X). Then, since x a M (X) by Proposition 2.7, we have that x a = y a for all a M (X). Similarly, we can prove the equivalences (i) (ii') (iii'). Proposition 2.12. Let X be a pseudo-BCI-algebra and let x X and a M (X). Then the following are equivalent: (i) x V (a), (ii) x b = a b for all b M (X), (iii) x ; b = a ; b for all b M (X). Proof. (i) (ii): Follows by Proposition 2.7. (ii) (i): Let x X and a M (X). Assume that x b = a b for all b M (X). Putting b = 1 we get x 1 = a 1. Hence, by Proposition 2.10, x and a are in the same branch of X, that is, x V (a). (i) (iii): Analogous. Let (X; , ;, 1) be a pseudo-BCI-algebra. Then X is p-semisimple if it satisfies for all x X, if x 1, then x = 1. Note that if X is a p-semisimple pseudo-BCI-algebra, then K(X) = {1}. Hence, if X is a p-semisimple pseudo-BCK-algebra, then X = {1}. Moreover, as it is proved in [3], M (X) is a p-semisimple pseudo-BCI-subalgebra of X and X is p-semisimple if and only if X = M (X). It is not difficult to see that the pseudo-BCI-algebras X, Y1 and Y from Examples 2.2, 2.3 and 2.5, respectively, are not p-semisimple, and the pseudo-BCI-algebra Y2 from Example 2.4 is a p-semisimple algebra. Proposition 2.13 ([3]). Let X be a pseudo-BCI-algebra. Then, for all a, b, x, y X, the following are equivalent: (i) X is p-semisimple, (ii) (x y) ; y = x = (x ; y) y, (iii) (x 1) ; 1 = x = (x ; 1) 1, (iv) if x a = x b, then a = b, (v) if x ; a = x ; b, then a = b, (vi) if a x = b x, then a = b, (vii) if a ; x = b ; x, then a = b. 3. Normal pseudo-BCI-algebras. A pseudo-BCI-algebra X is called normal if X = K(X) M (X). Otherwise, it is called non-normal. Remark. Every pseudo-BCK-algebra and every p-semisimple pseudo-BCIalgebra are normal. A pseudo-BCI-algebra X is called strongly normal if X is normal and K(X) = {1} and M (X) = {1}. Example 3.1. It is easy to see that the pseudo-BCI-algebra X from Example 2.2 is strongly normal, and the pseudo-BCI-algebra Y from Example 2.5 is non-normal. Theorem 3.2. Let X be a pseudo-BCI-algebra. Then the following are equivalent: (i) X is normal, (ii) ((x 1) 1) x {x, 1} for any x X, (iii) ((x 1) 1) ; x {x, 1} for any x X. Proof. (i) (ii): Let X be normal. Then X = K(X) M (X). Let x X. If x K(X), then ((x 1) 1) x = 1 x = x {x, 1}. If x M (X), then ((x 1) 1) x = x x = 1 {x, 1}. (ii) (i): Let ((x 1) 1) x {x, 1} for any x X. Take z X. If ((z 1) 1) z = z, then, by (b9), b(11) and (b12), z 1 = (((z 1) 1) z) 1 = (((z 1) 1) 1) ; (z 1) = (z 1) ; (z 1) =1 Hence, z 1, that is, z K(X). If ((z 1) 1) z = 1, then, (z 1) 1 z. Hence and by (a2) and (b12), z = (z 1) 1, which means that z M (X). Hence, X = K(X) M (X), that is, it is normal. (i) (iii): Analogously. In next theorem we construct some strongly normal pseudo-BCI-algebra. But first, we prove the following lemma. Lemma 3.3. Let X be a pseudo-BCI-algebra. Then (i) for any x X and y K(X), (x y) (x 1) = 1 = ((x 1) (x y)) 1, (x y) ; (x 1) = 1 = ((x 1) ; (x y)) 1, (x ; y) ; (x ; 1) = 1 = ((x ; 1) ; (x ; y)) 1, (x ; y) (x ; 1) = 1 = ((x ; 1) (x ; y)) 1, (ii) for any x K(X) and a M (X), x a = a = x ; a = (a x) 1 = (a ; x) 1, (iii) if X = K(X) M (X), then a x = a 1 = a ; x for any a M (X)\{1} and x K(X). Proof. (i) Let x X and y K(X). By (b1) and (b6), (x y) (x 1) = 1. Then, by (b10), 1 = (x y) (x 1) ((x 1) (x y)) 1. Hence, by (b1), (x y) (x 1) = 1 = ((x 1) (x y)) 1. Next, by (b4), (b11) and (b12) we have (x y) ; (x 1) = x ((x y) ; 1) = x ((x 1) ; (y 1)) = x ((x 1) ; 1) = (x 1) ; (x 1) = 1. Now, it is easy to see that (x y) ; (x 1) = 1 = ((x 1) ; (x y)) 1. Similarly, we can prove other equations of (i). (ii) Let x K(X) and a M (X). From Proposition 2.12 we immediately have that x a = a = x ; a. Moreover, by (b10) and (b12), a = x a ((a x) 1 and a = x ; a ((a ; x) 1. Since a M (X), we get (ii). (iii) Let X = K(X) M (X), a M (X)\{1} and x K(X). By (ii), (a x) 1 = a = 1. Hence, a x K(X), that is, a x M (X)\{1}. / Then, (a 1) (a x) M (X). But, by (i), (a x) (a 1) = 1 = ((a 1) (a x)) 1. Thus, a x a 1 and (a 1) (a x) = 1, that is, also a 1 a x. Therefore, a x = a 1. Similarly, we prove that a ; x = a 1. Remark. Note that the assumption X = K(X) M (X) in Lemma 3.3 (iii) is valid. Indeed, let Y be the pseudo-BCI-algebra from Example 2.5. We know that K(Y ) = {(x, 0, 0) : x 0} and M (Y ) = {(0, y, z) : y, z R}. Then for x < 0 and a1 , a2 R we have (0, a1 , a2 ) (x, 0, 0) = (0, a1 , a2 ) ; (x, 0, 0) = (x, -a1 , -a2 e-a1 ) = (0, a1 , a2 ) (0, 0, 0) = (0, -a1 , -a2 e-a1 ). Theorem 3.4. Let Y be a pseudo-BCK-algebra, Z be a (proper) p-semisimple pseudo-BCI-algebra and Y Z = {1}. Then there exists a unique pseudo-BCI-algebra X such that X = Y Z, K(X) = Y and M (X) = Z. Proof. First, the operations on Y and Z we denote by the same symbols and ;. Define on X = Y Z binary operations and as follows x y if x, y Y or x, y Z, y if x Y and y Z\{1}, xy= x 1 if x Z\{1} and y Y and x x ; y if x, y Y or x, y Z, y if x Y and y Z\{1}, y= x ; 1 if x Z\{1} and y Y. We show that (X; , , 1) is a pseudo-BCI-algebra. We check the conditions (i)(v) of Proposition 2.1. Since Y and Z are pseudo-BCI-algebras, we only need checking these conditions for the elements which are not all in Y and not all in Z. Particularly, (iii) and (iv) are satisfied. Now, we prove (v). Let x Y and y Z. Assume that x y = 1 = y x. Then, y = x y = 1. This means that x = 1 x = 1, that is, x = y = 1. Thus, (v) is satisfied. Next, we show the identity (i). Let x, x1 , x2 Y and y, y1 , y2 Z. Then (1) (x y1 ) [(y1 y2 ) (x y2 )] = y1 ; [(y1 y2 ) ; y2 ] = y1 ; y1 = 1, [(x y2 ) (y1 y2 )] = (y1 1) ; [y2 ; (y1 (2) (y1 x) y2 )] = (y1 1) ; (y1 1) = 1, (3) (y1 y2 ) [(y2 x) (y1 x)] = (y1 y2 ) ; [(y2 1) ; (y1 1)] = 1, [(x1 x2 ) (y x2 )] = (y 1) [(x1 x2 ) (4) (y x1 ) (y 1)] = (y 1) ; (y 1) = 1, [(y x2 ) (x1 x2 )] = y [(y 1) (x1 (5) (x1 y) [(y 1) ; 1] = y ; y = 1, x2 )] = y [(x2 y) (x1 y)] = (x1 x2 ) (y ; y) = (6) (x1 x2 ) y ; y = 1. Thus, (i) is also satisfied. Similarly we can prove (ii). Therefore, (X; , , 1) is a pseudo-BCI-algebra. Now, note that x 1 = x 1 for every x X. This means that x 1 = 1 if and only if x 1 = 1, and (x 1) 1 = x if and only if (x 1) 1 = x. Hence, K(X) = Y and M (X) = Z. Finally, we show uniqueness of pseudo-BCI-algebra (X; , , 1). Let (X; , , 1) be a pseudo-BCI-algebra such that X = Y Z, K(X) = Y and M (X) = Z. If x, y Y or x, y Z, then x y=xy=xy and x y=x;y=x y. If x Y and y Z\{1}, then, by Lemma 3.3, x y=y=xy and x y=y=x y. If x Z\{1} and y Y , then, again by Lemma 3.3, x and x Therefore, (X; , y=x 1=x;1=x , 1). y. y=x 1=x1=xy , 1) = (X; , Remark. Notice that a pseudo-BCI-algebra X constructed in Theorem 3.4 is strongly normal. Example 3.5. Take the following pseudo-BCK-algebra Y = {, , , 1} equipped with the operations and ; given by the following tables (see [6]): 1 1 1 1 1 1 1 1 1 1 1 1 ; 1 1 1 1 1 1 1 1 1 1 1 1 and the following p-semisimple pseudo-BCI-algebra Z = {a, b, c, d, e, 1} equipped with the operations and ; given by the following tables (see [4]): a b c d e 1 a 1 c e b d a b d 1 a e c b c e a 1 d b c d b e c 1 a d e c d b a 1 e 1 a b d c e 1 ; a b c d e 1 a 1 d b e c a b c 1 e a d b c b e 1 d a c d e a c 1 b d e d c a b 1 e 1 a b d c e 1 Then, using Theorem 3.4, we can construct the new pseudo-BCI-algebra (X; , , 1) such that X = Y Z and the operations and are as follows: a b c d e 1 1 1 1 a b c d e 1 1 1 a b c d e 1 1 a b c d e 1 a a a a 1 d e b c a b b b b c 1 a e d b c d d d e a 1 c b d d c c c b e d 1 a c e e e e d c b a 1 e 1 a b c d e 1 and a b c d e 1 1 1 1 a b c d e 1 1 1 a b c d e 1 1 a b c d e 1 a a a a 1 c b e d a b b b b d 1 e a c b c d d d b e 1 c a d d c c c e a d 1 b c e e e e c d a b 1 e 1 a b c d e 1 Obviously, K(X) = Y and M (X) = Z, that is, X is strongly normal. 4. Extensions of pseudo-BCI-algebras. Let X and X be pseudo-BCIalgebras. If X is a subalgebra of X , then X is called an extension of X. If X is p-semisimple (respectively, strongly normal, non-normal), then X is called a p-semisimple (respectively, strongly normal, non-normal) extension of X. If |X \X| = 1, then X is called a simple extension of X. First, we show some simple lemma. Consider the map p : X X such that p(x) = x 1 for all x X. Obviously, p(x) = x ; 1 for all x X. Note that Im(p) = M (X), Ker(p) = K(X) and if X is p-semisimple, then p is surjective. Lemma 4.1. Let X be a p-semisimple pseudo-BCI-algebra. Then, for all ; ; a X, maps fa , fa , ga , ga : X X such that fa (x) = x a, ; fa (x) = x ; a, ga (x) = a x, ; ga (x) = a ; x ; for all x X, are injective. Moreover, ga and ga are also surjective. ; Proof. Since X is p-semisimple, immediately by Proposition 2.13, fa , fa , ; ga , ga are injective. Moreover, for all x X, by (b4) we have ; (ga fa )(x) = ga (x ; a) = a (x ; a) = x ; (a a) = x ; 1 = p(x) and ; ; (ga fa )(x) = ga (x a) = a ; (x a) = x (a ; a) = x 1 = p(x) ; Hence, since p is surjective, maps ga and ga are surjective. ; ; Remark. Note that ga fa = ga fa and the map p can be decomposed into an injection and a bijection. Theorem 4.2. Let X be a p-semisimple pseudo-BCI-algebra. Then (i) there is no p-semisimple simple extension of X if |X| 2, (ii) there is a unique strongly normal simple extension of X, (iii) there is no non-normal simple extension of X. Proof. (i) Let X be a p-semisimple pseudo-BCI-algebra and |X| 2. Assume that X = X {x0 } is a p-semisimple extension of X. Since |X| 2, we can take x X\{1}. Now, take the map gx : X X . By Lemma 4.1 (X ) = X and g (X) = X. Note that g (x ) X. Inwe have gx 0 x x deed, if gx (x0 ) X \X = {x0 }, then x x0 = x0 = 1 x0 and by Proposition 2.13, x = 1, which is impossible. Hence, gx (x0 ) X. Thus, (X ) = g (X) {g (x )} = X and we have a contradiction. gx 0 x x (ii) First, there is a unique (pseudo-)BCK-algebra B0 = {0, 1} in which the operation is as follows 0 1 0 1 1 1 0 1 Now, it is sufficient to take a pseudo-BCI-algebra X = B0 X as in Theorem 3.4. Obviously, X is the unique strongly normal simple extension of X. (iii) It follows from (i) and the fact that for any pseudo-BCI-algebra Y we have K(Y ) = {1} if and only if M (Y ) = Y . Corollary 4.3. If X is a p-semisimple pseudo-BCI-algebra such that |X| 3, then X is not a simple extension of any pseudo-BCI-algebra. For arbitrary pseudo-BCI-algebras we have the following theorem. Theorem 4.4 ([4]). Any pseudo-BCI-algebra has a simple extension. Remark. Note that for a pseudo-BCI-algebra X a new element of its simple extension X constructed in [4] belongs to K(X). This means that if X is strongly normal (respectively, non-normal), then also X is strongly normal (respectively, non-normal).
Annales UMCS, Mathematica – de Gruyter
Published: Jun 1, 2015
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