@ Appl. Gen. Topol. 21, no. 2 (2020), 247-264 doi:10.4995/agt.2020.13049 c© AGT, UPV, 2020 Topological distances and geometry over the symmetrized Omega algebra Mesfer Alqahtani a, Cenap Özel a and Hanifa Zekraoui b a Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Ara- bia (mesfer ¯ alqhtani@hotmail.com,cozel@kau.edu.sa) b Department of Mathematics, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria (hzekraoui421@gmail.com) Communicated by F. Lin Abstract The aim of this paper is to study some topological distances properties, semidendrites and convexity on the symmetrized omega algebra. Fur- thermore, some properties and exponents on the symmetrized omega algebra are introduced. 2010 MSC: 15A80; 16Y60; 54F65. Keywords: omega algebra; symmetrized Omega algebra; semidendrite; ex- ponents; convex and topology. 1. Introduction Omega algebra unifies the min and the max plus algebras and introduces an original structure which in fact is an ”abstract tropical algebra”. termed it as ”Omega algebra” or in short just, ”ω− algebra”. The R−∞ and R∞ and their nearby structures, like min − max and max − times algebras, etc., are all subsumed under Omega algebra. All these are idempotent semirings which sometimes also termed as dioids. In the previous studies, for the construction of all such semirings, an ordered infinite abelian group is mandatory, see [4], [5], [7] and [8]. In ω− algebra, the definition is extended to cyclically ordered abelian groups and also for finite sets under some suitable ordering, for more details one can refer to [3]. Received 23 January 2020 – Accepted 29 July 2020 http://dx.doi.org/10.4995/agt.2020.13049 M. Alqahtani, C. Özel and H. Zekraoui The aim of this paper is to define some topological distances, semidendrites, convexity, some properties and exponents on the symmetrized Omega algebra. Our paper is organized as follows. In Section 2, we review some basic facts for Omega algebra and a brief of the symmetrized Omega algebra, furthermore the rules of calculation and the absolute value on the symmetrized Omega algebra. In Section 3, some properties and exponents on the symmetrized Omega algebra are introduced. We study some properties of the given topological distances and semidendrite on the symmetrized Omega algebra in Section 4. In Section 5, we generalize the notion of convex sets in paper [6] over Sω. This paper is produced from the PhD thesis of Mr. Mesfer Hayyan Alqahtani in King Abdulaziz University.. 2. Some preliminaries in abstract Omega algebra In this section, we recall some basic facts for Omega algebra, the sym- metrized Omega algebra, rules of calculation and ω−absolute value. For more details see [3]. 2.1. Omega algebra. Let (G, ◦, e) be an abelian group. Let A be a closed subset of G and e ∈ A. Then (A, ◦, e) is a submonoid of G. Assume that ω is an indeterminate (may belong to A or G, as we will see in Examples 1 and 2. Obviously, in this case ω is no longer an indeterminate). Because the terms are generated from tropical geometry, so such an indeterminate may be termed as a tropical indeterminate. Definition 2.1 ([3]). We say that Aω = A∪{ω} is an Omega algebra (in short ω− algebra) over the group G in case Aω is closed under two binary operations, ⊕, ⊗ : Aω × Aω −→ Aω, such that ∀a1, a2, a3 ∈ A, the following axioms are satisfied: 1) a1 ⊕ a2 = a1 or a2; 2) a1 ⊕ ω = a1 = ω ⊕ a1; 3) ω ⊕ ω = ω; 4) a1 ⊗ a2 = a2 ⊗ a1 ∈ A; 5) (a1 ⊗ a2) ⊗ a3 = a1 ⊗ (a2 ⊗ a3); 6) a1 ⊗ e = a1; 7) a1 ⊗ ω = ω ⊗ a1 = ! ω if ω ∕= e a1 if ω = e ; 8) ω ⊗ ω = ω; 9) a1 ⊗ (a2 ⊕ a3) = (a1 ⊗ a2) ⊕ (a1 ⊗ a3). Remark 2.2 ([3]). 1) ⊕ is a pairwise comparison operation, such as, max, min, inf, sup, up, down, lexicographic ordering, or any thing else that compairs two elements of Aω. Obviously, it is associative and commutative and the tropical indeterminate ω plays the role of the identity. Hence (Aω, ⊕, ω) is a commutative monoid. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 248 Topological distances and geometry over the symmetrized Omega algebra 2) ⊗ is also associative and commutative on Aω and e plays the role of the multiplicative identity of Aω. Hence (Aω, ⊗, e) is also a commutative monoid. 3) The left distributive law (9) also gives the right distributive law. 4) Every element of Aω is an idempotent under ⊕. 5) Altogether, we write both structures as: Aω = (Aω, ⊕, ⊗, ω, e). This is an idempotent semiring also called ”dioid” in literature. Remark 2.3 ([3]). In this note, we confined ourselves to only ω− algebras over abelian groups and rings. 2.2. The symmetrized Omega algebra. In this subsection, we give a brief of the symmetrized Omega algebra, for more details see [3]. Let (G, ◦, e) be an abelian group and (Aω, ⊕, ⊗, ω, e) an ω−algebra over the group G. We consider the set of ordered pairs Pω = A2ω. Let ≤ be the ordering defined on Aω by the relation (2.1) a ≤ b ⇐⇒ a ⊕ b = b which gives a total order on Aω and for all a ∈ Aω, we have ω ≤ a. For a ∕= b, such that a ⊕ b = b, we denote by a < b. Let ▽ be the relation defined on Pω as follows: for all (a, b), (c, d) ∈ Pω (a, b) ▽ (c, d) ⇐⇒ a ⊕ d = b ⊕ c. Definition 2.4 ([3]). Let ∼ be the equivalence relation close to ▽ defined as follows: for all (a, b), (c, d) ∈ Pω, (a, b) ∼ (c, d) ⇐⇒ ! (a, b) ▽ (c, d) if a ∕= b and c ∕= d (a, b) = (c, d) otherwise In addition to the class element ω = (ω, ω); for all a ∈ Aω, with a ∕= ω, we have three kinds of equivalence classes: (a) (a, ω) = {(a, b) ∈ Pω, b < a}, called positive ω−element. (b) (ω, a) = {(b, a) ∈ Pω, b < a}, called negative ω−element. (c) (a, a) called balenced ω−element. Proposition 2.5 ([3]). The addition operation ⊕ defined by (a, b)⊕(c, d) = (a ⊕ c, b ⊕ d) on the quotient set Pω∼ is well defined and satisfies the axioms (1), (2) and (3) of Definition 2.1. Proposition 2.6 ([3]). The set Pω∼ is closed under the binary multiplication operation ⊗ defined as follows: for all (a, b), (c, d) ∈ Pω∼ ; (a, b)⊗(c, d) = ((a ⊗ c) ⊕ (b ⊗ d), (a ⊗ d) ⊕ (b ⊗ c)) and satisfies axioms from (4) to (9) of Definition 2.1 with the unit class element e = (e, ω). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 249 M. Alqahtani, C. Özel and H. Zekraoui Definition 2.7 ([3]). The structure "Pω ∼ , ⊕, ⊗, ω, e # is called the symmetrized ω−algebra over the abelian group G × G and denoted by Sω. In the coming sections just for simplicity we will use ⊕, ⊗, ω and e instead of ⊕, ⊗, ω and e respectively. Remark 2.8 ([3]). 1) Despite the nature of the positive and the negative ω−elements, they are not the inverses of each others for the additive operation ⊕, 2) We have three symmetrized ω−subalgebras of Sω, S(+)ω = $ (a, ω), a ∈ Aω % , S(−)ω = $ (ω, a), a ∈ Aω % , S(0)ω = $ (a, a), a ∈ Aω % . 3) The three symmetrized ω−subalgebras of Sω are connected by the zero class element ω. 4) The positive ω−elements and the negative ω−elements are called signed and denoted by S∨ω = S (+) ω ∪ S (−) ω , where the zero class (ω, ω) corre- sponds to ω. 2.3. Rules of calculation in Omega. Let a ∈ Aω. Then we admit the fol- lowing notations: +a = (a, ω), − a = (ω, a), · a = (a, a). By results in Proposition 2.5, Proposition 2.6 and the above notation, it is easy to verify the rules of calculation in the following proposition: Proposition 2.9 ([3]). For all a, b ∈ Aω, we have (i) (+a) ⊕ (+b) = + (a ⊕ b) ; (ii) (+a) ⊕ (−b) = & ' ( +a if b < a −b if b > a ·a if b = a ; (iii) (±a) ⊕ (·b) = ! ±a if b < a ·b if b > a : (iv) (−a) ⊕ (−b) = − (a ⊕ b) ; (v) (+a) ⊗ (+b) = + (a ⊗ b) ; (vi) (+a) ⊗ (−b) = − (a ⊗ b) ; (vii) (±a) ⊗ (·b) = · (a ⊗ b) ; (viii) (−a) ⊗ (−b) = + (a ⊗ b) . From the previous rules, we can notice that the sign of the result in the ad- dition operation follows the greater element in Aω. While in the multiplication operation, the balance sign is the strong one (has priority). From Proposition 2.9, we can deduce the following : c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 250 Topological distances and geometry over the symmetrized Omega algebra Proposition 2.10 ([3]). The map |.|ω : Sω −→ Aω, such that for all a ∈ Aω, |+a|ω = |−a|ω = |·a|ω = a is an absolute value on Sω. We call it the ω−absolute value. 3. Some properties and exponents on Sω In this section, we give some properties and exponents on Sω. 3.1. Some properties on Sω. As Aω is isomorphic to the subdioid of pairs (a, ω), a ∈ Aω, then Aω itself can be considered as a subdioid of Pω. If u = (a, b) ∈ Sω, then we have two unary operators: ⊖ (the ω-algebraic minus operator) and (·)• (the balance operator) such that ⊖u = (b, a) and u• = u ⊕ (⊖u). Proposition 3.1. For all u = (a, b), v = (c, d) ∈ Sω, we have: (i) u• = (⊖u)• = (u•)•; (ii) u ⊗ v• = (u ⊗ v)• = u• ⊗ v = u• ⊗ v•; (iii) ⊖ (⊖u) = u, ⊖ (u ⊕ v) = (⊖u) ⊕ (⊖v) and (⊖u) ⊗ v = ⊖ (u ⊗ v) = u ⊗ (⊖v). Proof. A direct calculations gives the desired results. □ Proposition 3.2. Let a, b ∈ Aω be arbitrary. Then 1) ⊖(+a) = −a; 2) ⊖(−a) = +a; 3) ⊖(·a) = ·a; 4) +a ⊖ +b = (a, b). Proof. 1) ⊖(+a) = ⊖(a, ω) = ⊖ (a, ω) = (ω, a) = −a; 2) ⊖(−a) = ⊖(ω, a) = ⊖ (ω, a) = (a, ω) = +a; 3) ⊖(·a) = ⊖(a, a) = ⊖ (a, a) = (a, a) = ·a; 4) +a ⊖ +b = (a, ω) ⊖ (b, ω) = (a, ω) ⊕ (⊖(b, ω)) = (a, ω) ⊕ (ω, b) = (a, b). □ One can easily prove that the ordering ≤ defined on Aω by the relation (3.1) a ≤ b ⇐⇒ a ⊕ b = b is a total order on Aω and for all a ∈ Aω, we have ω ≤ a. For a ∕= b, such that a ⊕ b = b, we denote by a < b. By the construction of the symmetrized Omega algebra, it is easy to extend the total order ≤ on Aω to the ordering: Definition 3.3. For any u = (a, b), v = (c, d) ∈ Sω, we have u ≤ v ⇐⇒ u ⊕ v = v Corollary 3.4. For any (a, b) ∈ Sω, we have (ω, ω) ≤ (a, b). Proof. (a, b) ⊕ (ω, ω) = (a ⊕ ω, b ⊕ ω) = (a, b) ⇐⇒ ω = (ω, ω) ≤ (a, b). □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 251 M. Alqahtani, C. Özel and H. Zekraoui Proposition 3.5. Let (a, b) ∈ Sω, where a, b ∈ Aω, then we have: 1) If a > b then (a, b) = (a, ω). 2) If a < b then (a, b) = (ω, b). 3) If a = b then (a, b) = (a, a) or (a, b) = (b, b). Proof. 1) Let a > b (a ∕= ω, because ω > b impossible). Since (a, ω) = {(a, b) ∈ Pω, b < a}, then (a, b) ∈ (a, ω). Also (a, b) ∈ (a, b), because (a, b) ∼ (a, b). Then we have (a, b) ∈ (a, ω)∩(a, b). Hence (a, b) = (a, ω), because any two equivalence classes are disjoint or equal. 2) Let a < b (b ∕= ω, because a < ω impossible). Since (ω, b) = {(a, b) ∈ Pω, b > a}, then (a, b) ∈ (ω, b). Also (a, b) ∈ (a, b), because (a, b) ∼ (a, b). Then we have (a, b) ∈ (ω, b) ∩ (a, b). Hence (a, b) = (ω, b), because any two equivalence classes are disjoint or equal. 3) If a = b, then we have (a, b) = (a, a) or (a, b) = (b, b). □ Proposition 3.6. Let u = (a, ω), v = (ω, b), z = (c, c) ∈ Sω, where a, b, c ∈ Aω are arbitrary. Then we have 1) If a ≥ b, then u ⊕ v = u ⊕ (⊖v); 2) If a < b, then u ⊕ (⊖v) = (⊖u) ⊕ (⊖v); 3) If b ≤ c, then v ⊕ z = (⊖v) ⊕ z = v ⊕ (⊖z) = (⊖v) ⊕ (⊖z); 4) If b > c, then (⊖v) ⊕ (⊖z) = (⊖v) ⊕ z and v ⊕ z = v ⊕ (⊖z). Proof. 1) (3.2) u ⊕ v = (a, ω) ⊕ (ω, b) = (a ⊕ ω, ω ⊕ b) = (a, b) = (a, ω) and (3.3) u ⊕ (⊖v) = (a, ω) ⊕ (b, ω) = (a ⊕ b, ω) = (a, ω) By Equations 3.2 and 3.3, we get u ⊕ v = u ⊕ (⊖u). 2) (3.4) u ⊕ (⊖u) = (a, ω) ⊕ (b, ω) = (a ⊕ b, ω) = (b, ω) and (3.5) (⊖u) ⊕ (⊖v) = (ω, a) ⊕ (b, ω) = (b, a) = (b, ω) By Equations 3.4 and 3.5, we get u ⊕ (⊖v) = (⊖u) ⊕ (⊖v). 3) (3.6) v ⊕ z = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, c) (3.7) (⊖v) ⊕ z = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (c, c) (3.8) v ⊕ (⊖z) = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, c) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 252 Topological distances and geometry over the symmetrized Omega algebra (3.9) (⊖v) ⊕ (⊖z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (c, c) By Equations 3.6, 3.7, 3.8 and 3.9, we get v⊕z = (⊖v)⊕z = v⊕(⊖z) = (⊖v) ⊕ (⊖z). 4) (3.10) (⊖v) ⊕ (⊖z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (b, c) = (b, ω) and (3.11) (⊖v) ⊕ z) = (b, ω) ⊕ (c, c) = (b ⊕ c, c) = (b, c) = (b, ω) By Equations 3.10 and 3.11, we get (⊖v) ⊕ (⊖z) = (⊖v) ⊕ z. Also (3.12) v ⊕ z = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, b) = (ω, b) and (3.13) v ⊕ (⊖z) = (ω, b) ⊕ (c, c) = (c, b ⊕ c) = (c, b) = (ω, b) By Equations 3.12 and 3.13, we get v ⊕ z = v ⊕ (⊖z). □ 3.2. Exponents on Sω. In this subsection, we assume that ⊗|A = ◦. Let sign (.) ∈ {+, −, ·}. Definition 3.7. By Proposition 2.9, we can define the ω−power of an element in Sω as following: Let sign(a)a ∈ Sω be arbitrary, where a ∈ Aω and n ∈ Z+, then (1): If a ∕= ω. (sign(a)a)⊗n = (sign(a)a) ⊗ ... ⊗ (sign(a)a) ) *+ , n times = sign(.)(a ⊗ ... ⊗ a) ) *+ , n times = sign(.)(a ◦ ... ◦ a) ) *+ , n times , where sign (.) = ! sign(a) if sign(a) ∈ {+, ·} or sign(a) = − and n is odd + if sign(a) = − and n is even ; (2): If a = ω, then (sign(ω)ω)⊗n = sign(ω)ω. Example 3.8. Let −a ∈ Sω, where a ∕= ω, then c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 253 M. Alqahtani, C. Özel and H. Zekraoui 1) (−a)⊗2 = (ω, a) ⊗2 = (ω, a) ⊗ (ω, a) =((ω ⊗ ω) ⊕ (a ⊗ a), (ω ⊗ a) ⊕ (a ⊗ ω)) = (a ⊗ a, ω) = (a ◦ a, ω) = + (a ◦ a) 2) (−a)⊗3 = (ω, a) ⊗3 =(ω, a) ⊗ (ω, a) ⊗ (ω, a) = ((ω ⊗ ω) ⊕ (a ⊗ a), (ω ⊗ a) ⊕ (a ⊗ ω)) ⊗ (ω, a) = (a ⊗ a, ω) ⊗ (ω, a) = ((a ⊗ a ⊗ ω) ⊕ (ω ⊗ a), (a ⊗ a ⊗ a) ⊕ (ω ⊗ ω)) = (ω, a ⊗ a ⊗ a) = (ω, a ◦ a ◦ a) = − (a ◦ a ◦ a) Theorem 3.9. Let Aω = (Aω, ⊕, ⊗, ω, e) be an ω− algebra over an abelian group G = (G, ◦, e) and Sω = " Aω×Aω ∼ , ⊕, ⊗, ω, e # is the symmetrized ω−algebra over the abelian group G×G. Let ⊗|A = ◦, ω ∕= e and a ∕= ω. Then the following are equivalent: 1) If a ∈ (A, ◦, e), has an inverse in A, then 2) a ∈ (Aω \ {ω}, ⊗, e) has a multiplicativ inverse, and then 3) +a, −a ∈ Sω \ S (0) ω have a multiplicative inverse, but ·a ∈ S (0) ω has no multiplicative inverse. Proof. (1 ⇒ 2) Let a ∈ (A, ◦, e) be arbitrary, which has an inverse and denoted by a−1, then a ⊗ a−1 = a ◦ a−1 = e. Hence a⊗−1 = a−1 is the multiplicative inverse of a in (Aω \ {ω}, ⊗, e). (2 ⇒ 3) Let a ∈ Aω be arbitrary, where a ∕= ω and ω ∕= e, which has a multiplicative inverse and denoted by a⊗−1, then (+a) ⊗ (+a⊗−1) = (a, ω) ⊗ (a⊗−1, ω) =((a ⊗ a⊗−1) ⊕ (ω ⊗ ω), (a ⊗ ω) ⊕ (ω ⊗ a⊗−1)) = (a ⊗ a⊗−1, ω) = (a ◦ a⊗−1, ω) = (e, ω) = e then +a⊗−1 is a multiplicative inverse of +a in Sω and (−a) ⊗ (−a⊗−1) = (ω, a) ⊗ (ω, a⊗−1) =((ω ⊗ ω) ⊕ (a ⊗ a⊗−1), (ω ⊗ a⊗−1) ⊕ (a ⊗ ω)) = (a ⊗ a⊗−1, ω) = (a ◦ a⊗−1, ω) = (e, ω) = e c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 254 Topological distances and geometry over the symmetrized Omega algebra then −a⊗−1 is a multiplicative inverse of −a in Sω. On the other hand suppose that ·a ∈ S(0)ω has a multiplicative inverse (x, y), where x, y ∈ Aω, then (·a) ⊗ (x, y) = (a, a) ⊗ (x, y) =((a ⊗ x) ⊕ (a ⊗ y), (a ⊗ y) ⊕ (a ⊗ x)) = ((a ◦ x) ⊕ (a ◦ y), (a ◦ y) ⊕ (a ◦ x)) = (e, ω) hence (a ◦ x) ⊕ (a ◦ y) = e and (a ◦ y) ⊕ (a ◦ x) = ω, thus contradiction (note that if (x, y) = ·a⊗−1, then we have (a ◦ a⊗−1) ⊕ (a ◦ a⊗−1) = e and (a ◦ a⊗−1) ⊕ (a ◦ a⊗−1) = ω, thus condradiction). (3 ⇒ 1) Let +a ∈ Sω be arbitrary, where a ∕= ω, ω ∕= e and the multiplica- tive inverse of +a in Sω is (x, y), where x, y ∈ Aω, then we have: (+a) ⊗ (x, y) = (a, ω) ⊗ (x, y) =((a ⊗ x) ⊕ (ω ⊗ y), (a ⊗ y) ⊕ (ω ⊗ x)) = (a ⊗ x, a ⊗ y) = (a ◦ x, a ◦ y) = (e, ω) = e then a ◦ x = e and a ◦ y = ω. Hence x = a−1 is the multiplicative of a in (A, ◦, e). Let −a ∈ Sω be arbitrary, where a ∕= ω, ω ∕= e and the multiplicative inverse of −a is (x, y), where x, y ∈ Aω, then we have: (−a) ⊗ (x, y) = (ω, a) ⊗ (x, y) =((ω ⊗ x) ⊕ (a ⊗ y), (ω ⊗ y) ⊕ (a ⊗ x)) = (a ⊗ y, a ⊗ x) = (a ◦ y, a ◦ x) = (e, ω) = e then a ◦ y = e and a ◦ x = ω. Hence y = a−1 is the multiplicative inverse of a in (A, ◦, e). □ Corollary 3.10. (Aω \ {ω}, ⊗, e) is a group if and only if for any +a or −a ∈ Sω \ S (0) ω , where ⊗|A = ◦, ω ∕= e and a ∕= ω has a multiplicative inverse. Theorem 3.11. In Sω, with n, m ∈ Z+, a ∈ Aω be arbitrary and ⊗|A = ◦, the following exponents hold: 1) (+a)⊗n ⊗ (+a)⊗m = (+a)⊗(m+n) 2) (·a)⊗n ⊗ (·a)⊗m = (·a)⊗(m+n) 3) (−a)⊗n ⊗ (−a)⊗m = (−a)⊗(m+n) 4) ((+a)⊗n)⊗m = (+a)⊗(m×n) 5) ((·a)⊗n)⊗m = (·a)⊗(m×n) 6) ((−a)⊗n)⊗m = (−a)⊗(m×n) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 255 M. Alqahtani, C. Özel and H. Zekraoui Proof. 1) Let a ∕= ω, then (+a)⊗n ⊗ (+a)⊗m = +(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (+a)⊗(m+n). If a = ω, then (+ω)⊗n ⊗ (+ω)⊗m = +ω ⊗ (+ω) = +ω = (+ω)⊗(m+n). 2) By direct calculation similar above, we obtain to the desired result. 3) We have three cases: Case I) Let a ∕= ω and n, m are odd (note that n + m is even), then (−a)⊗n ⊗ (−a)⊗m = −(a ◦ ... ◦ a) ) *+ , n times ⊗ −(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) If a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = −ω ⊗ (−ω) = +ω = (−ω)⊗(m+n). Case II) Let a ∕= ω and n, m are even (note that n + m is even), then (−a)⊗n ⊗ (−a)⊗m = +(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = +((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 256 Topological distances and geometry over the symmetrized Omega algebra If a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = +ω ⊗ (+ω) = +ω = (−ω)⊗(m+n). Case III) Let a ∕= ω, n is odd and m is even (note that n + m is odd and we get to the same result if n is even and m is odd), then (−a)⊗n ⊗ (−a)⊗m = −(a ◦ ... ◦ a) ) *+ , n times ⊗ +(a ◦ ... ◦ a) ) *+ , m times = −((a ◦ ... ◦ a) ) *+ , n times ⊗ (a ◦ ... ◦ a) ) *+ , m times ) = −((a ◦ ... ◦ a) ) *+ , n times ◦ (a ◦ ... ◦ a) ) *+ , m times ) = −(a ◦ ... ◦ a) ) *+ , n+m times = (−a)⊗(m+n) If a = ω, then (−ω)⊗n ⊗ (−ω)⊗m = −ω ⊗ (+ω) = −ω = (−ω)⊗(m+n). 4) Let a ∕= ω, then ((+a)⊗n)⊗m = (+(a ◦ ... ◦ a) ) *+ , n times )⊗m = (+(a ◦ ... ◦ a) ) *+ , n times ⊗ ... ⊗ + (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ⊗ ... ⊗ (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +((a ◦ ... ◦ a) ) *+ , n times ◦ ... ◦ (a ◦ ... ◦ a) ) *+ , n times) *+ , m times ) = +(a ◦ ... ◦ a) ) *+ , n×m times = (+a)⊗(m×n) If a = ω, then ((+ω)⊗n)⊗m = (+ω)⊗m = +ω = (+ω)⊗(m×n). 5) By direct calculation similar above, we obtain to the desired result. 6) By direct calculation similar above with respect to n and m which is even or odd, we obtain to the desired result. □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 257 M. Alqahtani, C. Özel and H. Zekraoui 4. Some topological distances properties on Sω By the Euclidean and frobenius distances definitions on papers [6] and [3] respectively, we give some topological distances properties over Sω. We denote the Euclidean metric in Rn by de,n or by de when it clear what n is. The Euclidean metric in Cn is the same Euclidean metric in R2n. As we have already mentioned, Rn will be usually considered with its Euclidean metric de,n and with the natural topology induced by de,n. From Paper [3] we recall that, some metrics were first time introduced and some algebraic and topological properties were studied in the symmetrized max-plus-algebra [6]. Let (G, ◦) be a finitely generated (or finite) abelian group and Aω be an ω−algebra over G such that the restriction ⊗|A = ◦. By the fundamental theorem of finitely generated (or finite) abelian groups, the group G is a direct sum (direct product) of its cyclic groups, which make it isomorphic to a direct sum of finite copies of the cyclic groups Z of integers with finite summands of quotients of Z or isomorphic to a direct sum of cyclic groups of the quotients Zqi of Z (in other words, according to the group is infinite or finite, there exist some natural numbers m, r, q1, ...,qr such that G ∼= Zm or G ∼= Zm ⊕Zq1 ⊕...⊕Zqr or G ∼= Zq1 ⊕...⊕Zqr ). In all cases we can represent an element of G by n-tuple of elements of Z for some natural number n via that isomorphism. From this point of view, we will define metrics (or semimetrics) on our ω−algebra via the distance in Z. Let Φ be a such isomorphism. For any a ∈ G, there exists n ∈ N and there exist α1, . . ., αn ∈ Z (also they can be the representatives of classes in Z), such that Φ (a) = (α1, ..., αn). Let us extend Φ on Aω as follows: (4.1) Φ (a) = ! (α1, ..., αn) for a ∈ A, n ∈ N 0 for a = ω 4.1. The Euclidean distance on Sω. In this section, we suppose that the positive integer n is the number of the direct sum (direct product) of copies of the cyclic groups Z of integers and/or the direct sum of the cyclic groups of quotients of Z, which is isomorphic to G, where G be a finitely generated (or finite) abelian group and Aω be an ω−algebra over G such that the restriction ⊗|A = ◦. Definition 4.1. The embedding of Sω into Cn is the mapping ϕ : Sω → Cn defined in the following way: Let θ be a cube root of unity, say, θ = −1+ √ 3i 2 . As elements of Sω are defined by three signs, then we can emerge from Sω into Cn by the map ϕ defined by: for all a ∈ A, and by (4.1) we have Φ (a) = (α1, α2, ..., αn), such that n ∈ N and α1, α2, ..., αn ∈ Z, then c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 258 Topological distances and geometry over the symmetrized Omega algebra ϕ (sign(a)a) = & --' --( θ (exp(α1), ..., exp(αn)), if a ∈ A and sign(a) = +, θ2 (exp(α1), ..., exp(αn)), if a ∈ A and sign(a) = −, (exp(α1), ..., exp(αn)), if a ∈ A and sign(a) = ·, (0, ..., 0) , if a = ω. We shall consider the natural metric in Sω that is defined below: Definition 4.2. Let d1 : Sω × Sω−→R be defined as follows: d1(sign (a) a, sign (b) b) = de,2n(ϕ(sign (a) a), ϕ(sign (b) b)) for all sign(a)a, sign(b)b ∈ Sω, where de,2n is the Euclidean distance of R2n. The metric d1 in Sω will be called the Euclidean distance in Sω. Remark 4.3. (1) 1) It is clear that the Euclidean distances in Sω is induce the natural topology of Sω. As a topological space, we shall consider Sω only with its natural topology. 2) We have (Cn, de,2n) is a metric space and let ϕ(Sω) ⊆ Cn. then the sub- space metric on ϕ(Sω) is defined by simply restricting the metric on Cn to points in ϕ(Sω). In other words, for any ϕ(sign(a)a), ϕ(sign(b)b) ∈ ϕ(Sω), we define ρ(ϕ(sign(a)a), ϕ(sign(b)b)) = de,2n(ϕ(sign(a)a), ϕ(sign(b)b)). Hence (ϕ(Sω), ρ) is a metric subspace of a metric space (Cn, de,2n). Proposition 4.4. Let (Sω, d1) and (ϕ(Sω), ρ) be two metrics spaces, then 1) The mapping ϕ is an isometry from (Sω, d1) onto (ϕ(Sω), ρ). 2) (Sω, d1) is homeomorphic to (ϕ(Sω), ρ). Proof. 1) Let sign(a)a, sign(b)b ∈ Sω be arbitrary, then ρ(ϕ(sign(a)a), ϕ(sign(b)b)) = de,2n(ϕ(sign (a) a), ϕ(sign (b) b)) = d1(sign (a) a, sign (b) b). Hence ϕ is an isometry map. 2) Since the map ϕ from (Sω, d1) onto (ϕ(Sω), ρ) is an isometry, then ϕ is homeomorphism map. Hence (Sω, d1) ∼= (ϕ(Sω), ρ). □ Corollary 4.5. Since the mapping ϕ from (Sω, d1) onto (ϕ(Sω), ρ) is an isom- etry, then (Sω, τd1) ∼= (ϕ(Sω), τρ) , where τd1 and τρ are the natural topologies induced by the usual metrics d1 and ρ respectively. Let X be a topological space. We recall that it is said that a point c ∈ X disconnects X between points a, b ∈ X if there exists a pair U, V of disjoint open sets in X such that X \ {c} = U ∪ V and a ∈ U, while b ∈ V . The space X is a dendrite if it is a continuum such that, for each pair a, b of distinct points of X, there exists a point c ∈ X which disconnects X between points a and b or the space X is continuum if X is compact, connected and metric c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 259 M. Alqahtani, C. Özel and H. Zekraoui space. It is said that X is a semicontinuum if, for each pair a, b of points of X, there is a continuum C in X such that a, b ∈ C. Let us say that X is a semidendrite if it is a semicontinuum. Proposition 4.6. The topological space (Sω, τd1) is a semidendrite. Proof. Since ϕ(Sω) ⊂ Cn, then we have ϕ(Sω) is a semidendrite. By Corollary 4.5, we have (Sω, τd1) is a semidendrite. □ 4.2. The Frobenius distance on Sω. We suppose that the group G is a direct sum of its n cyclic subgroups (then G is a finitely generated group). Then for every a ∈ A, we have Φ (a) = (α1, α2, ..., αn), such that n ∈ N. Let ca be a right circulant matrix, such that the first row is (α1, α2..., αn), then ca = . /////// 0 α1 α2 . . . αn−1 αn αn α1 . . . αn−2 αn−1 αn−1 αn . . . αn−3 αn−2 ... ... ... ... ... α3 α4 . . . α1 α2 α2 α3 . . . αn α1 1 2222222 3 Each row of this matrix is right cyclic shift of the row above it. As right circulant matrices are diagonalizable in a same basis, we can benefit from this property to define a metric on Sω by using Frobenius norm ‖‖F (it is a matrix norm). Let θ be a cube root of unity, say, θ = −1+ √ 3i 2 . As elements of Sω are defined by three signs, then we can emerge from Sω into the algebra of right circulant matrices by the map φ defined by: for all a ∈ Aω, φ (sign(a)a) = & --' --( θ exp ((ca)) , if a ∈ A and sign(a) = +, θ2 exp ((ca)) , if a ∈ A and sign(a) = −, exp ((ca)) , if a ∈ A and sign(a) = ·, (0)n×n , if a = ω, Where (0)n×n is the zero right circulant matrix. Let us recall the Frobenius norm, sometimes also called the Euclidean norm is a matrix norm of an m × n matrix A, defined as the square root of the sum of the absolute squares of its elements, ‖A‖F = 4556 m7 i=1 n7 j=1 | aij |2 Let Cn×n(C), be the space of all right circulant matrices over C, the definition of distance over Cn×n(C) give by dF (A, B) = ‖A − B‖F , where A, B ∈ Cn×n(C). Hence (Cn×n(C), dF ) is a metric space. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 260 Topological distances and geometry over the symmetrized Omega algebra Corollary 4.7. The exponential function of right circulant matrix is right circulant matrix [1] and the multiplication scalar of right circulant matrix is right circulant matrix. Hence, we have φ(Sω) ⊂ Cn×n(C). Corollary 4.8. The map φ : Sω → Cn×n(C) is an embedding. Definition 4.9. Let d2 : Sω × Sω −→ R be defined by d2 (sign(a)a, sign(b)b) = dF (φ (sign(a)a) , φ (sign(b)b)) = ‖φ (sign(a)a) − φ (sign(b)b)‖F for all sign(a)a, sign(b)b ∈ Sω, where dF is the Frobenius distance of the right circulant matrices Cn×n(C). The metric d2 in Sω will be called the Omega Frobenius distance in Sω. Remark 4.10. Let (Cn×n(C), dF ) be a metric space. let φ(Sω) be a non-empty subset of the right circulant matrices space Cn×n(C). Then dF |φ(Sω)×φ(Sω), the restriction of dF to φ(Sω) × φ(Sω) is a metric on φ(Sω). Let us denoted for dF |φ(Sω)×φ(Sω) by ρ. In other words, for any φ(sign(a)a), φ(sign(b)b) ∈ φ(Sω), we define ρ(φ(sign(a)a), φ(sign(b)b)) = dF (φ(sign(a)a), φ(sign(b)b)). Hence (φ(Sω), ρ) is a metric subspace of a metric space (Cn×n(C), dF ). Proposition 4.11. Let (Sω, d2) and (φ(Sω), ρ) be two metrics spaces, then 1) The mapping φ is an isometry from (Sω, d2) onto (φ(Sω), ρ). 2) (Sω, d2) is homeomorphic to (φ(Sω), ρ). Proof. 1) Let sign(a)a, sign(b)b ∈ Sω be arbitrary, then ρ(φ(sign(a)a), φ(sign(b)b)) = dF (φ(sign(a)a), φ(sign(b)b)) = d2(sign (a) a, sign (b) b). Hence φ is an isometry map. 2) Since the map φ from (Sω, d2) onto (φ(Sω), ρ) is an isometry, then φ is homeomorphism map. Hence (Sω, d2) ∼= (φ(Sω), ρ). □ Corollary 4.12. Since the mapping φ from (Sω, d2) onto (φ(Sω), ρ) is an isom- etry, then (Sω, τd2) ∼= (φ(Sω), τρ) , where τd2 and τρ are the natural topologies induced by the Omega Frobenius distance d2 and Frobenius distance ρ respec- tively. Proposition 4.13. Let (Sω, τd2) and (Cn×n(C), τdF ) be a topological space, then φ : (Sω, τd2) → (Cn×n(C), τdF ) is an embedding map. Proof. By Corollary 4.12, we have the space Sω is homeomorphic with the subspace φ(Sω) of Cn×n(C). Hence the map φ is embedding. □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 261 M. Alqahtani, C. Özel and H. Zekraoui 5. Convexity in symmetrized Omega algebra In this section, we generalize the notion of convex sets in paper [6] over Sω. We suppose that n ∈ Z+ is the number of the direct sum (direct product) of copies of the cyclic groups Z of integers and/or the direct sum of the cyclic groups of quotients of Z, which is isomorphic to G, where G be a finitely generated (or finite) abelian group and Aω be an ω−algebra over G such that the restriction ⊗|A = ◦ and ω /∈ A. By ω− absolute value, we have the following corollaries: Corollary 5.1. Each sign(a)a ∈ Sω can be written as sign(a)a = sign(a) |sign(a)a|ω where sign(a) ∈ {+. − .·}. Corollary 5.2. One can write the formulas for actions in Sω: (5.1) sign(a)a ⊕ sign(b)b = sign(a ⊕ b)( |sign(a)a|ω ⊕ |sign(b)b|ω) (5.2) sign(a)a ⊗ sign(b)b = (sign(a)) ⊗ (sign(b)) (|sign(a)a|ω ⊗ |sign(b)b|ω) where, sign(a ⊕ b) = & --' --( sign(a), a > b, sign(b), b > a, sign(a), a = b, sign(a) = sign(b), · , a = b, sign(a) ∕= sign(b), and the multiplication table for signs as follow. (sign(a))⊗(sign(b)) = & ' ( +, if sign(a) = sign(b) = + or−, −, if sign(a) ∕= sign(b), sign(a) ∕= · and sign(b) ∕= ·, ·, if either sign(a) = · or sign(b) = ·, For any sign(a)a, sign(b)b ∈ Sω, we can define sign(a)a ⊕ sign(b)b ∈ Sω and sign(a)a ⊗ sign(b)b ∈ Sω as in Equations 5.1 and 5.2 respectively. Then (Sω, ⊕, ⊗) is a semiring. By a scalar we mean an element of Aω. For sign(a)a ∈ Sω, µ ∈ Aω, we define µ ⊗ sign(a)a = sign(a)(µ ◦ |sign(a)a|ω).(5.3) The operation ⊗ : Aω × Sω → Sω is well defined, and we can consider it as an outer operation from Aω ×Sω to Sω or the restriction to Aω ×Sω of the inner operation ⊗ in Sω because Aω ⊂ Sω. Considering ⊗ as the outer operation, we can look at Sω as at a semimodule(”vector space over semiring”) over the semiring Aω. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 262 Topological distances and geometry over the symmetrized Omega algebra We recall that, for each pair of points x1, x2 of a space X, a segment [x1, x2] with end-points x1 and x2 is defined, then a set A ⊆ X is called convex if, for all x1, x2 ∈ A, we have [x1, x2] ⊆ A see [2]. In particular, for sign(a)a, sign(b)b ∈ Sω, we can consider distinct kinds of segments with end- points sign(a)a and sign(b)b, between them,traditional segments and semimod- ule segments. These kinds of segments in Sω lead to distinct kinds of convexity in Sω. Let us turn our attention to traditional and semimodule convexity in Sω. A convex set in Cn is a set B ⊆ Cn such that, for each pair x1, x2 ∈ B, the traditional segment [x1, x2] in the space Cn is contained in B. Definition 5.3. A set B ⊆ Sω will be called traditionally convex if ϕ(B) is convex in Cn. Remark 5.4. 1) Let us notice that a nonempty set B ⊆ Sω is simultaneously compact and traditionally convex in Sω if and only if B is a traditional segment in Sω. 2) All traditionally convex subsets of Sω are connected. 5.1. Semimodule convex sets in Sω. The semimodule structure of Sω al- lows us to think about semimodule convexity of subsets of Sω. In analogy to conventional algebra, we define a semimodule segment [sign(a)a, sign(b)b]s with end-points sign(a)a ∈ Sω and sign(b)b ∈ Sω as follows: [sign(a)a, sign(b)b]s ={(µ ⊗ sign(a)a) ⊕ (γ ⊗ sign(a)b) : µ, γ ∈ Aω, with µ ⊕ γ = e}. Definition 5.5. A set B ⊆ Sω is said to be semimodule convex if, for each pair sign(a)a, sign(b)b of points of B, the semimodule segment [sign(a)a, sign(b)b]s is contained in B. Acknowledgements. The authors would thank the referees for the valuable remarks and advices. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 263 M. Alqahtani, C. Özel and H. Zekraoui References [1] A. C. F. Bueno, On the exponential function of right circulant matrices, International Journal of Mathematics and Scientific Computing 3, no. 2 (2013). [2] L. Hörmander, Notions of convexity, Progress in Mathematics 127, Birkhäuser, Boston- Basel-Berlin (1994). [3] S. Khalid Nauman, C. Ozel and H. Zekraoui, Abstract Omega algebra that subsumes min and max plus algebras, Turkish Journal of Mathematics and Computer Science 11 (2019) 1–10. [4] G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal of Mathematical Sciences 140, no. 3 (2007), 426–444. [5] D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, 2015. [6] C. Ozel, A. Piekosz, E. Wajch and H. Zekraoui, The minimizing vector theorem in symmetrized max-plus algebra, Journal of Convex Analysis 26, no. 2 (2019), 661–686. [7] J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50–69, Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998. [8] I. Simon, Recognizable sets with multiplicities in the tropical semiring, in: Mathematical Foundations of Computer Science (Carlsbad, 1988), Lecture Notes in Computer Science, vol. 324, Springer, Berlin, 1988, pp. 107–120. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 264