EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1483-1496 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some Structural Properties of Fully UP-semigroups Dianne P. Gomisong1,2,∗, Rowena T. Isla1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for Graph Theory, Algebra and Analysis, Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. This paper investigates a new class of algebra related to UP-algebras and semigroups called fully UP-semigroups (or f -UP-semigroups). It establishes some structural properties of f - UP-semigroups. It also introduces and examines f -UP-fields, f -UP-domains, f -UP-ideals, and quotient f -UP-semigroups. Moreover, it investigates the relationship between an f -UP-field and an f -UP-domain. 2010 Mathematics Subject Classifications: 03G25, 08A99 Key Words and Phrases: UP-algebra, f -UP-semigroup, f -UP-field, f -UP-domain, f -UP-ideal, Quotient f -UP-semigroup 1. Introduction In 1966, Y. Imai and K. Iseki [5] introduced the idea of BCK-algebra as a generaliza- tion of the concept of set-theoretic difference and propositional calculi. In the same year, K. Iseki [6] introduced the notion of BCI-algebra as a generalization of BCK-algebra. Studies on different types of algebraic structures followed, among them B-algebras, G- algebras, BCH-algebras, BE-algebras, and SU-algebras. In 2009, C. Prabpayak and U. Leerawat [11] introduced the notion of KU-algebra and investigated some related proper- ties. In 2017, A. Iampan [3] introduced a class of algebra called UP-algebra (UP means the University of Phayao). He established its structure and defined some concepts such as UP-subalgebras, UP-ideals, congruences, and UP-homomorphism. He determined some properties of UP-homomorphism, which led to four isomorphism theorems for UP-algebras. He also presented some connections between UP-algebras and KU-algebras and showed that the notion of UP-algebra is a generalization of KU-algebra. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3527 Email addresses: dianne.gomisong@g.msuiit.edu.ph (D. Gomisong), rowena.isla@g.msuiit.edu.ph (R. Isla) http://www.ejpam.com 1483 c© 2019 EJPAM All rights reserved. D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1484 In 1993, Jun, Hong, and Roh [7] introduced a class of algebra related to BCI-algebras and semigroups with distributive laws property, called a BCI-semigroup. Jun et al. [8, 9] renamed the BCI-semigroup as the IS-algebra and studied related properties. In 2018, F. Kareem and E. Hasan [10] introduced the concept of KU-semigroup which is a combination of KU-algebra and semigroup. In the same year, A. Iampan [4] introduced a new class of algebra called a fully UP-semigroup (or f -UP-semigroup) which is a combination of UP-algebra and semigroup. In this study, the notion of f -UP-semigroup is investigated and some of its properties are established. 2. Preliminaries An algebra of type (2, 0) is an algebra with a binary operation and a constant element. Definition 1. [11] A KU-algebra is an algebra (X; ∗, 0) of type (2, 0) satisfying the fol- lowing axioms: for all x, y, z ∈ X, (KU1) (x ∗ y) ∗ [(y ∗ z) ∗ (x ∗ z)] = 0, (KU2) 0 ∗ x = x, (KU3) x ∗ 0 = 0, (KU4) x ∗ y = y ∗ x = 0 implies x = y. Example 1. [11] Let X = {0, a, b, c} be a set with a binary operation ∗ defined by the following Cayley table: ∗ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 0 0 0 Then, (X; ∗, 0) is a KU-algebra. Definition 2. [3] A UP-algebra is an algebra (X; ∗, 0) of type (2, 0) satisfying the following axioms: for all x, y, z ∈ X, (UP1) (y ∗ z) ∗ [(x ∗ y) ∗ (x ∗ z)] = 0, (UP2) 0 ∗ x = x, (UP3) x ∗ 0 = 0, (UP4) x ∗ y = y ∗ x = 0 implies x = y. Example 2. [3] Let X = {0, a, b, c} be a set with a binary operation ∗ defined by the following Cayley table: D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1485 ∗ 0 a b c 0 0 a b c a 0 0 b b b 0 a 0 b c 0 a 0 0 Then, (X; ∗, 0) is a UP-algebra. Definition 3. [3] Let X be a UP-algebra. A subset S of X is called a UP-subalgebra of X if the constant zero of X is in S and (S; ∗, 0) itself forms a UP-algebra. Definition 4. [1] Define x∧y = (y∗x)∗x. Then X is said to be a commutative UP-algebra if for any x, y ∈ X, (y ∗ x) ∗ x = (x ∗ y) ∗ y, that is, x ∧ y = y ∧ x. Definition 5. [3] Let X be a UP-algebra. Then, a subset I of X is called a UP-ideal of X if it satisfies: (i) the constant zero of X is in I, and (ii) for any x, y, z ∈ X, x ∗ (y ∗ z) ∈ I and y ∈ I imply x ∗ z ∈ I. Proposition 1. [3] In a UP-algebra (X; ∗, 0), the following properties hold: for any x, y, z ∈ X, (i) x ∗ x = 0, (ii) x ∗ y = 0 and y ∗ z = 0 imply x ∗ z = 0, (iii) x ∗ y = 0 implies (z ∗ x) ∗ (z ∗ y) = 0, (iv) x ∗ y = 0 implies (y ∗ z) ∗ (x ∗ z) = 0, (v) x ∗ (y ∗ x) = 0, (vi) (y ∗ x) ∗ x = 0 implies x = y ∗ x, and (vii) x ∗ (y ∗ y) = 0. The next result gives a relationship between UP-algebras and KU-algebras. Theorem 1. [3] Any KU-algebra is a UP-algebra. The converse of Theorem 1 does not hold. To see this, consider the UP-algebra (X; ∗, 0) in Example 2. Let x = 0, y = a, and z = c. Observe that (x ∗ y) ∗ [(y ∗ z) ∗ (x ∗ z)] = (0 ∗ a) ∗ [(a ∗ c) ∗ (0 ∗ c)] = a ∗ (b ∗ c) = a ∗ b = b 6= 0, so (KU1) is not satisfied. Thus, (X; ∗, 0) is not a KU-algebra. In view of Theorem 1, the notion of UP-algebras is a generalization of KU-algebras. Proposition 2. [3] A nonempty subset S of a UP-algebra (X; ∗, 0) is a UP-subalgebra of X if and only if S is closed under the ∗ operation. D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1486 Let X be a UP-algebra and A be a nonempty subset of X. Then X ∗ A is given by X ∗A = ⋃ x∈X,a∈A (x ∗ a). Theorem 2. [3] Let X be a UP-algebra and B a UP-ideal of X. Then X ∗ B ⊆ B. In particular, B is a UP-subalgebra of X. Let (X; ∗, 0) be a UP-algebra and B be a UP-ideal of X. Define the binary relation ∼B on X as follows: for all x, y ∈ X, x ∼B y if and only if x ∗ y ∈ B and y ∗ x ∈ B. An equivalence relation ρ on X is called a congruence if for any x, y, z ∈ X, xρy implies (x ∗ z)ρ(y ∗ z) and (z ∗ x)ρ(z ∗ y). If x ∈ X, then the ρ-class of x is [x]ρ defined as [x]ρ = {y ∈ X : yρx}. The set of all ρ-classes is called the quotient set of X by ρ, and is denoted by X/ρ. That is, X/ρ = {[x]ρ : x ∈ X}. Theorem 3. [3] Let (X; ∗, 0) be a UP-algebra and B a UP-ideal of X. Then the following hold: (i) the ∼B-class [0]∼B is a UP-ideal and a UP-subalgebra of X, (ii) a ∼B-class [x]∼B is a UP-ideal of X if and only if x ∈ B, (iii) a ∼B-class [x]∼B is a UP-subalgebra of X if and only if x ∈ B, and (iv) (X/ ∼B; ∗, [0]∼B ) is a UP-algebra under the operation ∗ defined by [x]∼B∗[y]∼B = [x∗ y]∼B for all x, y ∈ X, called the quotient UP-algebra of X induced by the congruence ∼B. Definition 6. [10] A KU-semigroup is a nonempty set X together with two binary oper- ations ∗ and · and a constant 0 satisfying the following: (KUS1) (X; ∗, 0) is a KU-algebra; (KUS2) (X, ·) is a semigroup; and (KUS3) the operation · is left and right distributive over the operation ∗, that is, x · (y ∗ z) = (x · y) ∗ (x · z) and (x ∗ y) · z = (x · z) ∗ (y · z). Example 3. [10] Let X = {0, a, b, c} be a set with the binary operations ∗ and · defined by the following Cayley tables: ∗ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 0 0 0 · 0 a b c 0 0 0 0 0 a 0 0 0 0 b 0 0 0 b c 0 0 b c D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1487 Then, (X; ∗, ·, 0) is a KU-semigroup. Definition 7. [4] A fully UP-semigroup (or f -UP-semigroup) is a nonempty set X to- gether with two binary operations ∗ and · and a constant 0 satisfying the following: (fUP1) (X; ∗, 0) is a UP-algebra; (fUP2) (X, ·) is a semigroup; and (fUP3) the operation · is left and right distributive over the operation ∗. A. Iampan [4] analogously introduced a left [resp., right ] UP-semigroup as a nonempty set X together with two binary operations ∗ and · and a constant 0 satisfying (fUP1), (fUP2), and the operation · is left [resp. right] distributive over the operation ∗. Thus, an f -UP-semigroup is both a left and a right UP-semigroup. Example 4. [4] Let X = {0, a, b, c} be a set with the binary operations ∗ and · defined by the following Cayley tables: ∗ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 · 0 a b c 0 0 0 0 0 a 0 0 0 0 b 0 0 0 a c 0 0 a 0 Then, (X; ∗, ·, 0) is an f -UP-semigroup. Example 5. Let X = {0, a, b, c} be a set with the binary operations ∗ and · defined by the following Cayley tables: ∗ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 0 0 0 · 0 a b c 0 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c 0 a b c Then, routine calculations show that (X; ∗, ·, 0) is an f -UP-semigroup. Example 6. Let X = {0, a, b, c, d} be a set with the binary operations ∗ and · defined by the following Cayley tables: ∗ 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 · 0 a b c 0 0 0 0 0 a 0 a b c b 0 b c a c 0 c a b D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1488 Then, routine calculations show that (X; ∗, ·, 0) is an f -UP-semigroup. Hereinafter, let X denote the f -UP-semigroup (X; ∗, ·, 0), unless otherwise indicated. Definition 8. A nonempty subset S of an f -UP-semigroupX is called an f -UP-subsemigroup of X if the constant 0 of X is in S and (S; ∗, ·, 0) itself forms an f -UP-semigroup. Obviously, {0} and X are f -UP-subsemigroups of X. In Example 4, the set S1 = {0, b} is an f -UP-subsemigroup of X, while the set S2 = {0, b, c} is not an f -UP-subsemigroup since b · c = a /∈ S2. The following remark immediately follows from Definitions 8, 7, and 3. Remark 1. Every f -UP-subsemigroup of (X; ∗, ·, 0) is a UP-subalgebra of X with respect to ∗. The converse of Remark 1 does not hold. To see this, consider Example 4. It can be easily verified that S = {0, b, c} is a UP-subalgebra of (X; ∗, 0) but S is not an f -UP- subsemigroup of (X; ∗, ·, 0) since b · c = a /∈ S. Definition 9. An f -UP-semigroup X is said to be commutative if a · b = b · a for all a, b ∈ X. If X is not commutative, then it is called a noncommutative f -UP-semigroup. Routine calculations show that the f -UP-semigroups in Examples 4 and 6 are commu- tative while the f -UP-semigroup in Example 5 is noncommutative since a·c = 0 6= a = c·a. Definition 10. Let X be an f -UP-semigroup. An element e ∈ X is called a unity in X if x · e = x = e · x for all x ∈ X. Proposition 3. Let X be an f -UP-semigroup. If the unity of X exists, then it is unique. Proof. Let X be an f -UP-semigroup with unity. Suppose 1, 1′ ∈ X both satisfy the properties of being a unity. Then, for all x ∈ X, x · 1 = 1 · x = x and x · 1′ = 1′ · x = x. If x = 1, we have 1 · 1′ = 1. If x = 1′, we have 1 · 1′ = 1′. Therefore, 1 = 1′. If an f -UP-semigroup X has unity, it shall be denoted by 1. Definition 11. Let X be an f -UP-semigroup with unity 1. An element a of X is called 1-invertible if there exists b ∈ X such that a · b = 1 = b · a. We next introduce the concepts of f -UP-field and f -UP-domain analogous to the definitions of JB-field and JB-domain given by J. Endam and J. Vilela [2]. Definition 12. Let X be an f -UP-semigroup with unity 1. Then X is called an f -UP-field if the following hold: (i) the semigroup (X, ·) is commutative; and (ii) every 0 6= a ∈ X is 1-invertible. D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1489 Definition 13. A nonzero element a of an f -UP-semigroup X is called a 0-divisor if there exists b ∈ X such that b 6= 0 and either a · b = 0 or b · a = 0. Note that 0 is not a 0-divisor. Remark 2. An element cannot be 1-invertible and a 0-divisor at the same time. Thus, an f -UP-field has no 0-divisors. Definition 14. Let X be an f -UP-semigroup with unity 1. Then X is called an f -UP- domain if the following hold: (i) the semigroup (X, ·) is commutative; and (ii) X has no 0-divisors. The f -UP-semigroup in Example 6 is an f -UP-domain. Remark 3. Every f -UP-field is an f -UP-domain. 3. Elementary Properties of f-UP-semigroups This section presents some elementary properties of f -UP-semigroups. Throughout this section, X means an f -UP-semigroup (X; ∗, ·, 0). Theorem 4. Let a, b, c ∈ X. Then the following properties hold: (i) a · 0 = 0 · a = 0, (ii) a · (0 ∗ b) = (0 ∗ a) · b = a · b, (iii) a · (b ∗ (0 ∗ c)) = (a · b) ∗ (a · c) and (b ∗ (0 ∗ c)) · a = (b · a) ∗ (c · a), (iv) a · (b ∧ c) = (a · b) ∧ (a · c) and (a ∧ b) · c = (a · c) ∧ (b · c), (v) If a · b = 0, then a · (b ∗ c) = a · c, (vi) If a · c = 0, then (a ∗ b) · c = b · c. Proof. Let a, b, c ∈ X. (i) By Proposition 1(i) and (fUP3), a · 0 = a · (0 ∗ 0) = (a · 0) ∗ (a · 0) = 0. Similarly, 0 · a = 0. (ii) By (UP2), a · (0 ∗ b) = a · b = (0 ∗ a) · b. (iii) By (UP2) and (fUP3), a · (b ∗ (0 ∗ c)) = a · (b ∗ c) = (a · b) ∗ (a · c). Similarly, (b ∗ (0 ∗ c)) · a = (b ∗ c) · a = (b · a) ∗ (c · a). (iv) By Definition 4 and (fUP3), a · (b ∧ c) = a · [(c ∗ b) ∗ b] = [a · (c ∗ b)] ∗ (a · b) = [(a ·c)∗(a ·b)]∗(a ·b) = (a ·b)∧(a ·c) and (a∧b) ·c = [(b∗a)∗a] ·c = [(b∗a) ·c]∗(a ·c) = [(b · c) ∗ (a · c)] ∗ (a · c) = (a · c) ∧ (b · c). D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1490 (v) Suppose a·b = 0. Then by (fUP3) and (UP2), a·(b∗c) = (a·b)∗(a·c) = 0∗(a·c) = a·c. (vi) If a ·c = 0, then by (fUP3) and (UP2), (a∗b) ·c = (a ·c)∗(b ·c) = 0∗(b ·c) = b ·c. The following theorem gives a necessary and sufficient condition for a subset of an f -UP-semigroup to be an f -UP-subsemigroup. Theorem 5. A nonempty subset S of an f -UP-semigroup (X; ∗, ·, 0) is an f -UP-subsemi- group of X if and only if x ∗ y, x · y ∈ S for all x, y ∈ S. Proof. Let ∅ 6= S ⊆ X. Suppose S is an f -UP-subsemigroup of X. Then by Defini- tion 8, (S; ∗, ·, 0) is an f -UP-semigroup. Thus, the binary operations ∗ and · are closed in S, that is, x∗y, x ·y ∈ S for all x, y ∈ S. Conversely, suppose x∗y, x ·y ∈ S for all x, y ∈ S. Then 0 = x ∗ x ∈ S. By Proposition 2, (S; ∗, 0) is a UP-subalgebra of X, hence (fUP1) holds. Let x, y, z ∈ S ⊆ X. Then x · y ∈ S by our assumption and x · (y · z) = (x · y) · z by associativity in X. Hence, (S, ·) is a semigroup and (fUP2) is satisfied. Moreover, (fUP3) holds for all x, y, z ∈ S ⊆ X. Thus, S is an f -UP-subsemigroup of X. Theorem 6. Let X be an f -UP-semigroup and {Ai : i ∈ I} a family of f -UP-subsemigroups of X. Then ⋂ i∈I Ai is an f -UP-subsemigroup of X. Proof. Since Ai is an f -UP-subsemigroup of X, 0 ∈ Ai for all i ∈ I. Thus, 0 ∈ ⋂ i∈I Ai and ⋂ i∈I Ai 6= ∅. Let x, y ∈ ⋂ i∈I Ai. Then for all i ∈ I, x, y ∈ Ai and by Theorem 5, x ∗ y, x · y ∈ Ai. Hence, x ∗ y, x · y ∈ ⋂ i∈I Ai. Therefore, ⋂ i∈I Ai is an f -UP-subsemigroup of X. The next result shows a relationship between KU-semigroups and f -UP-semigroups. Theorem 7. Any KU-semigroup is an f -UP-semigroup. Proof. Let X = (X; ∗, ·, 0) be a KU-semigroup. By Theorem 1, (X; ∗, 0) is a UP- algebra. By Definition 6, (X, ·) is a semigroup and left and right distributivity hold for · over ∗, thus X is an f -UP-semigroup. Remark 4. The converse of Theorem 7 does not hold. To see this, let X = {0, a, b, c, d} be a set with the binary operations ∗ and · defined by the following Cayley tables: ∗ 0 a b c d 0 0 a b c d a 0 0 0 0 0 b 0 b 0 0 0 c 0 b b 0 0 d 0 b b d 0 · 0 a b c d 0 0 0 0 0 0 a 0 0 0 0 0 b 0 0 0 0 0 c 0 0 0 0 0 d 0 0 0 0 0 D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1491 Then by routine calculations, (X; ∗, ·, 0) is an f -UP-semigroup. Let x = 0, y = c, and z = a. Observe that (x∗y)∗ [(y∗z)∗(x∗z)] = (0∗c)∗ [(c∗a)∗(0∗a)] = c∗(b∗a) = c∗b = b, so (KU1) is not satisfied. Thus, (X; ∗, ·, 0) is not a KU-semigroup. Theorem 8. Let X be an f -UP-semigroup with unity 1 and let T be the set of all 1- invertible elements of X. Then (i) 1 ∈ T , (ii) 0 /∈ T , and (iii) a · b ∈ T for all a, b ∈ T . Proof. Let T be the set of all 1-invertible elements of X. (i) Since 1 · 1 = 1, 1 ∈ T . Thus, T 6= ∅. (ii) Suppose 0 ∈ T . Then there exists b ∈ X such that 0 · b = 1 = b · 0. But 0 · b = 0 and so, 0 = 1, a contradiction. Thus, 0 /∈ T . (iii) Let a, b ∈ T . Then there exist c, d ∈ X such that a · c = 1 = c · a and b · d = 1 = d · b. Moreover, d · c ∈ X. By (fUP2), (a · b) · (d · c) = ((a · b) · d) · c = (a · (b · d)) · c = (a ·1) ·c = a ·c = 1 and (d ·c) ·(a ·b) = ((d ·c) ·a) ·b = (d ·(c ·a)) ·b = (d ·1) ·b = d ·b = 1. Hence, a · b ∈ T . The next result establishes a relation between 0-divisors and the cancellation property of an f -UP-semigroup. Theorem 9. If an f -UP-semigroup X has no 0-divisors, then left and right cancellation laws hold, that is, for all a, b, c ∈ X, a 6= 0, a · b = a · c implies b = c (left cancellation) and b · a = c · a implies b = c (right cancellation). If either left or right cancellation law holds, then X has no 0-divisors. Proof. Let a, b, c ∈ X such that a ·b = a ·c and a 6= 0. Then a ·(b∗c) = (a ·b)∗(a ·c) = 0 by Proposition 1(i). Since X has no 0-divisors and a 6= 0, we have b∗c = 0. Since a·b = a·c, we have 0 = a · (b ∗ c) = (a · b) ∗ (a · c) = (a · c) ∗ (a · b) = a · (c ∗ b) and so, c ∗ b = 0. By (UP4), b = c. Hence, the left cancellation law holds. Similarly, the right cancellation law holds. Conversely, suppose one of the cancellation laws holds, say, the left cancellation. Let a be a nonzero element of X and b ∈ X. Suppose a ·b = 0. Then by Theorem 4(i), a ·b = a ·0 and so by left cancellation, b = 0. Suppose b · a = 0 and b 6= 0. Then by Theorem 4(i), b · a = b · 0 and so by left cancellation, a = 0, a contradiction. Therefore, b = 0 and X has no 0-divisors. Similarly, the right cancellation law implies that X has no 0-divisors. Theorem 10. A finite commutative f -UP-semigroup X with more than one element and without 0-divisors is an f -UP-field. D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1492 Proof. Let a1, a2, . . . , an be the distinct elements of X. Let a ∈ X with a 6= 0. Now, a · ai ∈ X for all i = 1, 2, . . . , n and so {a · a1, a · a2, . . . , a · an} ⊆ X. If a · ai = a · aj , then by Theorem 9, ai = aj . Thus, the elements a · a1, a · a2, . . . , a · an are distinct and so X = {a · a1, a · a2, . . . , a · an}. Hence, one of the elements, say a · ai, must be equal to a. Since X is commutative, ai ·a = a ·ai = a. Let b ∈ X. Then there exists aj ∈ X such that b = a ·aj . Thus, b ·ai = ai · b = ai · (a ·aj) = (ai ·a) ·aj = a ·aj = b. This implies that ai is the unity of X. We denote the unity of X by 1. Now, 1 ∈ X = {a ·a1, a ·a2, . . . , a ·an} and so one of the elements, say a ·ak, must be equal to 1. By commutativity, ak ·a = a ·ak = 1. Hence, every nonzero element of X is 1-invertible. Therefore, X is an f -UP-field. As a consequence of Theorem 10, the following corollary holds. Corollary 1. Every finite f -UP-domain is an f -UP-field. 4. f-UP-Ideal and the Quotient f-UP-semigroup Definition 15. A nonempty subset I of an f -UP-semigroup X is called an f -UP-ideal of X if the following hold: (fUPI1) the constant 0 of X is in I, (fUPI2) for any x, y, z ∈ X, x ∗ (y ∗ z) ∈ I and y ∈ I imply x ∗ z ∈ I, and (fUPI3) for any a ∈ I, x ∈ X, a · x, x · a ∈ I. Obviously, the subsets {0} and X are f -UP-ideals of X. Consider the f -UP-semigroup in Example 4. Routine calculations show that the set I1 = {0, a, b} is an f -UP-ideal of X while the set I2 = {0, b, c} is not an f -UP-ideal of X since b · c = a /∈ I2. Theorem 11. Let (X; ∗, ·, 0) be an f -UP-semigroup and I an f -UP-ideal of X. Then I is an f -UP-subsemigroup of X. Proof. By (fUP1), (X; ∗, 0) is a UP-algebra and by definition, I is a UP-ideal of the UP-algebra X. By Theorem 2, I is a UP-subalgebra of X. Let x, y ∈ I ⊆ X. Then by Proposition 2, x ∗ y ∈ I. Since I is an f -UP-ideal of the f -UP-semigroup X, x · y ∈ I by (fUPI3). Thus, I is an f -UP-subsemigroup of X by Theorem 5. Theorem 12. Let X be an f -UP-semigroup and {Ai : i ∈ I } be a nonempty collection of f -UP-ideals of X. Then ⋂ i∈I Ai is an f -UP-ideal of X. Proof. Suppose {Ai : i ∈ I } is a nonempty collection of f -UP-ideals of X. Since 0 ∈ Ai for all i ∈ I , 0 ∈ ⋂ i∈I Ai and so ⋂ i∈I Ai 6= ∅. Suppose x, y, z ∈ X such that x ∗ (y ∗ z) ∈ ⋂ i∈I Ai and y ∈ ⋂ i∈I Ai. Then x ∗ (y ∗ z) ∈ Ai and y ∈ Ai for all i ∈ I . Since D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1493 each Ai is an f -UP-ideal for all i ∈ I , it follows that x ∗ z ∈ Ai for all i ∈ I . Hence, x ∗ z ∈ ⋂ i∈I Ai. Let a ∈ ⋂ i∈I Ai and x ∈ X. Then a ∈ Ai for all i ∈ I . Since each Ai is an f -UP-ideal for all i ∈ I , a · x, x · a ∈ Ai for all i ∈ I . Hence, a · x, x · a ∈ ⋂ i∈I Ai. Therefore, ⋂ i∈I Ai is an f -UP-ideal of X. Let (X; ∗, ·, 0) be an f -UP-semigroup and I be an f -UP-ideal of X. Define the binary relation ∼I on X as follows: for all x, y ∈ X, x ∼I y if and only if x ∗ y ∈ I and y ∗ x ∈ I. Denote [x]I as the equivalence class containing x ∈ X and X/I as the set of all equivalence classes of X with respect to “∼I”, that is, [x]I = {y ∈ X : x ∼I y} and X/I = {[x]I : x ∈ X}. Remark 5. Let X be an f -UP-semigroup and I be an f -UP-ideal of X. Then x ∈ [x]I for all x ∈ X. Lemma 1. Let X be an f -UP-semigroup and I be an f -UP-ideal of X. Then [x]I = [y]I if and only if x ∼I y. Proof. Suppose [x]I = [y]I . Since y ∈ [y]I = [x]I , we have x ∼I y. Conversely, suppose x ∼I y. Let z ∈ [x]I . Then x ∼I z. By symmetric property, z ∼I x. By transitivity, z ∼I y and by symmetric property, y ∼I z and so, z ∈ [y]I . Thus, [x]I ⊆ [y]I . Let z ∈ [y]I . Then y ∼I z. By transitivity, x ∼I z, that is, z ∈ [x]I . Thus, [y]I ⊆ [x]I . Hence, [x]I = [y]I . Proposition 4. Let X be an f -UP-semigroup and I be an f -UP-ideal of X. Then (i) [0]I = I, (ii) [x]I = I if and only if x ∈ I, for all x ∈ I, and (iii) I ∗ [x]I = [x]I for all x ∈ X. Proof. Let I be an f -UP-ideal of X. (i) If x ∈ [0]I , then by definition, 0 ∼I x and by (UP2), x = 0 ∗ x ∈ I. Thus, [0]I ⊆ I. Let x ∈ I. By (UP2), 0 ∗ x = x ∈ I. By (UP3) and (fUPI1), x ∗ 0 = 0 ∈ I. Thus, 0 ∼I x and so, x ∈ [0]I . Hence, I ⊆ [0]I . Therefore, [0]I = I. (ii) Suppose [x]I = I. Then by Remark 5, x ∈ I. Conversely, let x ∈ I. By (UP2), 0∗x = x ∈ I. By (UP3) and (fUPI1), x∗0 = 0 ∈ I. Thus, 0 ∼I x, and by Lemma 1, [0]I = [x]I . By (i), I = [x]I . (iii) For all x ∈ X, [x]I = [0 ∗ x]I = [0]I ∗ [x]I as defined in Theorem 3(iv). By (i), [x]I = I ∗ [x]I . D.Gomisong, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1483-1496 1494 Theorem 13. If X is an f -UP-semigroup and I an f -UP-ideal of X, then (X/I; ∗, ·, [0]I) is an f -UP-semigroup, where ∗ and · are defined by [x]I∗[y]I = [x∗y]I and [x]I ·[y]I = [x·y]I , respectively. If X is commutative, then X/I is commutative and if X has unity, then X/I has unity. Proof. Let I be an f -UP-ideal of X. Then I is a UP-ideal of the UP-algebra (X; ∗, 0). By Theorem 3, (X/I; ∗, [0]I) is a UP-algebra, where ∗ is defined by [x]I ∗ [y]I = [x∗y]I . We show that the binary operation · on X/I is well-defined. Let [x]I = [x′]I and [y]I = [y′]I . Then x ∼I x′ and y ∼I y′ which imply x ∗ x′, x′ ∗ x, y ∗ y′, y′ ∗ y ∈ I. By Theorem 4(iii), (UP2), and (fUPI3), (x ·y)∗ (x ·y′) = x · (y ∗ (0∗y′)) = x · (y ∗y′) ∈ I and (x ·y′)∗ (x ·y) = x·(y′∗(0∗y)) = x·(y′∗y) ∈ I. Thus, x·y ∼I x·y′. Similarly, (x·y′)∗(x′·y′) = (x∗(0∗x′))·y′ = (x∗x′) ·y′ ∈ I and (x′ ·y′)∗ (x ·y′) = (x′ ∗ (0∗x)) ·y′ = (x′ ∗x) ·y′ ∈ I. Thus, x ·y′ ∼I x′ ·y′. By transitivity, x · y ∼I x′ · y′. By Lemma 1, [x]I · [y]I = [x · y]I = [x′ · y′]I = [x′]I · [y′]I . Let [x]I , [y]I , [z]I ∈ X/I. Since (X, ·) is a semigroup, then [x]I · ([y]I · [z]I) = [x]I · [y · z]I = [x · (y · z)]I = [(x · y) · z]I = [x · y]I · [z]I = ([x]I · [y]I) · [z]I . Hence, (X/I, ·) is semigroup. Moreover, by distributive property on X, [x]I · ([y]I ∗ [z]I) = [x]I · [y ∗ z]I = [x · (y ∗ z)]I = [(x · y) ∗ (x · z)]I = [x · y]I ∗ [x · z]I = ([x]I · [y]I) ∗ ([x]I · [z]I) and ([x]I ∗ [y]I) · [z]I = [x ∗ y]I · [z]I = [(x ∗ y) · z]I = [(x · z) ∗ (y · z)]I = [x · z]I ∗ [y · z]I = ([x]I · [z]I) ∗ ([y]I · [z]I). Thus, the distributive property holds on X/I. Therefore, (X/I; ∗, ·, [0]I) is an f -UP- semigroup. Suppose X is commutative. Then x · y = y · x for all x, y ∈ X. Let [x]I , [y]I ∈ X/I. Then [x]I · [y]I = [x · y]I = [y ·x]I = [y]I · [x]I . Hence, X/I is commutative. If X has unity 1, then X/I has unity [1]I since [x]I · [1]I = [x ·1]I = [x]I and [1]I · [x]I = [1 ·x]I = [x]I for any x ∈ X. The f -UP-semigroup (X/I; ∗, ·, [0]I) in Theorem 13 is called the quotient f -UP-semigroup of X by I. REFERENCES 1495 5. Conclusion This paper investigated fully UP-semigroups, a new class of algebra related to UP- algebras and semigroups, which was introduced by A. Iampan [4] in 2018. 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