@ Appl. Gen. Topol. 23, no. 1 (2022), 201-212 doi:10.4995/agt.2022.14778 © AGT, UPV, 2022 Topological Krasner hyperrings with special emphasis on isomorphism theorems Manoranjan Singha and Kousik Das Department of Mathematics, University of North Bengal, Darjeeling-734013, India (manoranjan.math@nbu.ac.in, das.kousik1991@nbu.ac.in) Communicated by A. Tamariz-Mascarúa Abstract Krasner hyperring is studied in topological flavor. It is seen that Kras- ner hyperring endowed with topology, when the topology is compatible with the hyperoperations in some sense, fruits new results comprising algebraic as well as topological properties such as topological isomor- phism theorems. 2020 MSC: 16Y20; 22A30. Keywords: topological hyperring; quotient hyperring; topological isomor- phism. 1. Introduction and relevant literature The theory of hyperalgebra which is extended in this article gets birth in the year 1934 but it gets acquaintance during the last two decades and so far it is wide in various branches of Mathematics including Physics and Chemistry: geometry [25, 26], graph theory [5, 27], codes [31, 10], cryptography [4], prob- ability [20], automata [19], artificial intelligence [17], lattice theory [14, 15], chemistry [1, 12], physics [8, 21], and all credits for these go to the hyperoper- ations. It is investigated that how the difference between hyperoperation and binary operation affects on the theory of topological Krasner hyperring, espe- cially on the topological isomorphisms. Hyperring, introduced by Krasner [16] is one of the most general structures so far in the literature that satisfies the ring-like axioms. Later, many mathematicians, like Ameri [3, 2], Massouros Received 11 December 2020 – Accepted 10 December 2021 http://dx.doi.org/10.4995/agt.2022.14778 https://orcid.org/0000-0003-4875-4330 https://orcid.org/0000-0001-8011-3635 M. Singha and K. Das [18], Spartalis [29], Davvaz [7], Stratigopoulos [30], Kemprasit [24] extended this field of study. In literature, a topological ring is a combination of two structures, namely a topological space and a ring. These two structures are connected in such a way that one affects another. In this paper, we generalize this concept as topological Krasner hyperring, supported by illustrative exam- ples. We also present an example that makes difference between the classical and the new concept. In the later part, we use the notion of complete parts to study isomorphism theorems on hyperrings. Let’s begin with some basic definitions and results which will be used as ready references in the sequel. On a nonempty set H, a hyperoperation is a function + : H × H → P∗(H), where P∗(H) is the collection of nonempty subsets of H. For nonempty subsets A,B of H and x ∈H, consider A + B = ⋃ a∈A, b∈B a + b, x + A = {x} + A and A + x = A + {x}. A Krasner hyperring is an algebraic structure (H, +, ·) satisfying the following axioms: (1) (H, +) is a canonical hypergroup, i.e., + is a hyperoperation on H such that (a) for every x,y,z ∈H, x + (y + z) = (x + y) + z, (b) for every x,y ∈H, x + y = y + x, (c) there exists 0 ∈H such that 0 + x = {x} for every x ∈H, (d) for every x ∈H, there exists a unique x′ ∈H such that 0 ∈ x+x′, (write −x instead of such x′), (e) z ∈ x + y implies y ∈−x + z and x ∈ z −y; (2) (H, ·) is a semigroup having zero as a bilaterally absorbing element, i.e., 0 ·x = x · 0 = 0 for all x ∈H. (3) The multiplication, ‘·’ is distributive with respect to the hyperoperation +. Throughout this context, hyperring stands for Krasner hyperring. The following elementary facts are the consequences of the above axioms: −(−x) = x, for any nonempty subset X of H, −X = {−x : x ∈ X} and −(x+y) = −x−y. Also, for all a,b,c,d ∈H, (a+b)·(c+d) ⊆ a·c+b·c+a·d+b·d. A nonempty subset K of the hyperring H is said to be a subhyperring of H if (K, +, ·) is itself a hyperring. The subhyperring K is a hyperideal of H if h ·k ∈ K and k ·h ∈ K for all h ∈ H and k ∈ K. The subhyperring K is said to be normal in H if and only if h + K−h ⊆ K for all h ∈ H. For a normal hyperideal K of a hyperring H the following results hold: (1) x + K = K + x for all x ∈H, (2) (x + K) + (y + K) = x + y + K for all x,y ∈H, (3) if x,y ∈H, x + y + K = z + K for all z ∈ x + y, (4) x + K = y + K for all y ∈ x + K. Let K1,K2 be two hyperideals of a hyperring H such that K2 is normal in H. Then, (1) K1 ∩K2 is a normal hyperideal of K1, © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 202 Topological Krasner hyperrings with special emphasis on isomorphism theorems (2) K2 is a normal hyperideal of K1 + K2. For a normal hyperideal K of a hyperring H, define an equivalence relation K∗ as follows: x ≡ y(modK) if and only if (x−y) ∩K 6= φ. Then, for all x ∈H, K∗(x) = x+K. The collection [H : K∗] = {K∗(x) : x ∈H} of all equivalence classes forms a hyperring together with the hyperoperations ⊕ and multiplication � defined as follows: K∗(x) ⊕K∗(y) = {K∗(z) : z ∈K∗(x) + K∗(y)}, K∗(x) �K∗(y) = K∗(x ·y). A homomorphism from a hyperring (H, +, ·) into another hyperring (H′, +′, ·′) is a map f : H→H′ such that f(x+y) ⊆ f(x)+′f(y) and f(x·y) = f(x)·′f(y), for all x,y ∈ H. A homomorphism f from (H, +, ·) into (H′, +′, ·′) is said to be a good homomorphism if f(x + y) = f(x) +′ f(y), for all x,y ∈ H. An onto homomorphism is called epimorphism. An isomorphism from (H, +, ·) onto (H′, +′, ·′) is a bijective good homomorphism and if such map exists, then write H ∼= H′. If f is an isomorphism from H onto H′, then f−1 is an iso- morphism from H′ onto H. For a homomorphism f : H → H′, the kernel of the homomorphism is defined as ker f = {x ∈ H : f(x) = 0H′}. It is seen (Example 1.2 [24]) that the kernel of a homomorphism may be empty, but, if it is nonempty (i.e., ker f 6= φ), then the following results ([24]) hold: (1) f(0H) = 0H′ ; (2) f(−x) = −f(x), for all x ∈H; (3) ker f is a hyperideal of H; (4) If f is injective, then ker f = {0H}; (5) If f is a good homomorphism and ker f = {0H}, then f is injective; (6) If f is a good homomorphism, f(H) is a subhyperring of H′. Note that, if f is onto, then f(x) = 0H′ for some x ∈H, i.e., ker f 6= φ. A nonempty subset C of a hyperring H is said to be a complete part of H if for any nonzero natural number n and for all x1,x2, · · · ,xn of H, the following implication holds: C ∩ n Σ i=1 xi 6= φ ⇒ n Σ i=1 xi ⊆ C. Let A and B be two nonempty subsets of the hyperring H such that A is a complete part of H and x ∈H. Then, (1) −x + x + A = x−x + A = A; (2) −A is a complete part of H; (3) x + A and A + x are complete parts; (4) B ⊆−x + A if and only if x + B ⊆ A. For more details about hyperring we refer to [16, 9, 24, 7]. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 203 M. Singha and K. Das 2. Topological Hyperring and Isomorphism Theorems To define topological hyperring, the codomain of the hyperoperation is to be topologized, but there is no straightforward way to obtain such topology. So, let’s consider the following. Lemma 2.1 ([13]). For a topological space (H,τ), the family B consisting of the sets SV = {U ∈ P∗(H) : U ⊆ V}, where V ∈ τ is a base for a topology τ∗ on P∗(H). Now, we are in a situation to define topological hyperring. Definition 2.2. Let (H, +, ·) be a Krasner hyperring endowed with some topol- ogy τ. Then, H is said to be a topological Krasner hyperring, denoted by (H, +, ·,τ), if with respect to the product topology on H×H and the topology τ∗ on P∗(H), the following maps (TH1) (x,y) 7→ x + y from H×H to P∗(H); (TH2) x 7→−x from H to H; (TH3) (x,y) 7→ x ·y from H×H to H; are continuous. For the ease of writing throughout this context, let’s write topological hyperring instead of topological Krasner hyperring. Every topological ring is a topological hyperring. Here, we consider some other examples. Example 2.3. Consider the hyperring (R, +, ·), where R = {0, 1, 2}, the hy- peroperation + and the binary operation ‘·’ are defined as follows + 0 1 2 0 {0} {1} {2} 1 {1} {0,2} {1} 2 {2} {1} {0} · 0 1 2 0 0 0 0 1 0 1 2 2 0 2 0 Let R be endowed with the topology τ = {φ,{1},{0, 2},R}. Then, (R, +, ·,τ) is a topological hyperring. Example 2.4. Consider the unit interval [0, 1] as a subspace of R with standard topology. For x,y ∈ [0, 1], let + be the hyperoperation defined as follows x + y = { {max{x,y}}, if x 6= y; [0,x], if x = y. Then ([0, 1], +, ·) is a topological hyperring, where ‘·’ is the usual multiplication on R. Remark 2.5. Unlike in topological rings, some results may fail to hold in the new setting. For, in the above Example 2.4, 1 2 ∈ [0, 1] and [0, 1 2 ) is open in [0, 1], but 1 2 ⊕ [0, 1 2 ) = {1 2 }, which is not open in [0, 1]. Lemma 2.6. In a topological hyperring (H, +, ·,τ), the following results hold. (1) For a ∈H, the map Ta(x) = a + x from H to P∗(H) is continuous. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 204 Topological Krasner hyperrings with special emphasis on isomorphism theorems (2) Let U be open and a complete part of H. Then, for a ∈H, a + U is an open subset of H. Moreover, if A is any subset of H, then A + U is an open subset of H. (3) The map I(x) = −x from H to H is a homeomorphism. Proof. (1) Being a restriction of the map + : H×H→P∗(H), Ta is continuous for any a ∈H. (2) Consider the basic open set SU of P∗(H) and a ∈H. Then, T−1−a (SU ) = {x ∈H : T−a(x) ∈ SU} = {x ∈H : −a + x ⊆ U} Suppose S = {x ∈ H : −a + x ⊆ U}. Take y ∈ S, then −a + y ⊆ U. So, y ∈ 0 + y ⊆ (a + (−a)) + y ⊆ a + (−a + y) ⊆ a + U. Again, if z ∈ a + U, then −a + z ⊆ −a + a + U = U, which implies z ∈ S. Hence, T−1−a (SU ) = a + U, which is open in H. For A ⊆H, A + U = ⋃ a∈A (a + U), which is also open for being arbitrary union of open sets. (3) The inverse of an element in the canonical hypergroup (H, +) is unique. So, the map I on H is a homeomorphism. � Remark 2.7. The open subsets in the above Example 2.3 are complete parts. Theorem 2.8. In a topological hyperring (H, +, ·,τ), the following straightfor- ward results easily hold. (1) For a neighborhood V of zero, there exist a neighborhood U of zero and a neighborhood W of x, where x ∈ H such that U + U ⊆ V and U ·W ⊆ V . (2) If V is any neighborhood of zero, then −V is also a neighborhood of zero. (3) Every neighborhood U of zero contains a symmetric neighborhood of zero (i.e., U ∩ (−U) ). (4) If U is a neighborhood of zero and n > 1, then there exists a symmetric neighborhood V of zero such that V + V + · · · + V︸ ︷︷ ︸ n terms ⊆ U. Proof. The proofs are straightforward. � Any subhyperring of a topological hyperring is also a topological hyperring when considering the relative topology on it, such subhyperrings are called topological subhyperrings. Let I be a hyperideal of a hyperring (H, +, ·) and H/I = {x + I : x ∈ H}. Then, (H/I,⊕,�) is a hyperring, called quotient hyperring of H by I, where (x + I) ⊕ (y + I) = {z + I : z ∈ x + y} and (x + I) � (y + I) = (x ·y) + I for x,y ∈H. Let I be a normal hyperideal of a topological hyperring (H, +, ·,τ) and ΦI be the canonical map of H onto H/I, defined by ΦI(x) = x + I for x ∈ H. Let’s topologize H/I by declaring the map ΦI to be quotient, i.e., a subset A © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 205 M. Singha and K. Das of H/I is open in H/I if and only if Φ−1I (A) is open in H. This topology is called the quotient topology on H/I and denoted by τΦ. Theorem 2.9. Let I be a normal hyperideal of a topological hyperring (H, +, ·,τ) such that the members of τ are complete parts and ΦI be the above mentioned map, then the following results hold. (1) ΦI is continuous, open good epimorphism. (2) (H/I,⊕,�,τΦ) is a topological hyperring. (3) If V is a fundamental system of neighborhoods of zero (i.e., 0) in H, then {ΦI(V ) : V ∈ V} is a fundamental system of neighborhoods of zero (i.e., 0 + I = I) for the quotient topology τΦ of H/I. Proof. (1) For x,y ∈H, ΦI(x + y) = {z +I : z ∈ x + y} = (x +I)⊕(y +I) = ΦI(x)⊕ΦI(y). ΦI is continuous as per the definition of the quotient topology. Let O be an open subset of H. Claim that Φ−1I (ΦI(O)) = O + I. To prove O + I ⊆ Φ−1I (ΦI(O)), take y ∈ O + I. Then, y ∈ p + I for some p ∈ O, which implies y + I = p + I. Thus, y ∈ Φ−1I (ΦI(O)). Now, take x ∈ Φ −1 I (ΦI(O)), then x + I ∈ ΦI(O), which implies x + I = r + I for some r ∈ O. Thus, x ∈ r + I ⊆ O + I. Hence, ΦI is an open map (by (2) of Lemma 2.6). (2) ΦI×ΦI is the map from H×H to H/I×H/I defined by (ΦI×ΦI)(x,y) = (ΦI(x), ΦI(y)) for all (x,y) ∈ H×H. As ΦI is a continuous open surjection, so is ΦI × ΦI. Then, ⊕◦ (ΦI × ΦI) = ΦI ◦ f and �◦ (ΦI × ΦI) = ΦI ◦ f, where f = + and · respectively. So, the continuity of f implies both ⊕ and � are continuous (by Theorem 5.3 of [33, p. 33]). Let I : H/I →H/I be defined by I(x + I) = (−x) + I for x ∈H. Also, I ◦ ΦI = ΦI ◦− is continuous, as − is continuous; hence I is continuous (by Theorem 5.3 of [33, p. 33]). (3) For every neighborhood U of zero in H/I, Φ−1I (U) is a neighborhood of zero in H, so there exists V ∈ V such that V ⊆ Φ−1I (U). Then, ΦI(V ) ⊆ ΦI(Φ −1 I (U)) = U. � Remark 2.10. It is clear from the above Theorem 2.9 that if A is some open subset of H/I, then there exists open subset A of H such that A = A/I. Theorem 2.11. Let B be a subhyperring and I be a normal hyperideal of a topological hyperring (H, +, ·,τ) such that I ⊆ B. If the members of τ are complete parts, then the quotient topology of B/I is identical with the topology induced on the subhyperring B/I of H/I by the quotient topology of H/I. Proof. Let ΦB,I : B →B/I and ΦH,I : H→H/I be the canonical surjections. Let O be open for the quotient topology of B/I. Then, Φ−1B,I(O) is open in B and Φ−1B,I(O) = B∩Q for some open subset Q of H. Claim that O = B/I∩ΦH,I(Q). For, clearly O ⊆ B/I ∩ ΦH,I(Q). For the converse, take α ∈ B/I ∩ ΦH,I(Q). Then, α = b + I for some b ∈B and α = q + I for some q ∈Q, which implies q ∈ b + I ⊆ B + I = B as I ⊆ B. Consequently, q ∈ B∩Q = Φ−1B,I(O), so α = q + I ∈ O. Now suppose R be open in B/I for the topology on B/I induced by the quotient topology on H/I. Then, R = B/I ∩ S for some open subset S of © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 206 Topological Krasner hyperrings with special emphasis on isomorphism theorems H/I. So, Φ−1B,I(R) = B∩ Φ −1 H,I(S), which is open in B. Hence, R is open for the quotient topology of B/I. � Corollary 2.12. Let B be a subhyperring and I be a normal hyperideal of a topological hyperring (H, +, ·,τ). If the members of τ are complete parts, then the quotient topology on (B +I)/I is identical with the topology induced by the quotient topology of H/I. Let’s define topological isomorphism and prove some topological isomor- phism theorems. Definition 2.13. A homomorphism f between two topological hyperrings sat- isfying ker f 6= φ is said to be a topological homomorphism if it is a continuous open mapping. If f is a good, one to one and onto topological homomorphism, then it is called a topological isomorphism and in this case the hyperrings are topologically isomorphic. Example 2.14. Consider the topological hyperring ([0, 1], +, ·,τu) as in Exam- ple 2.4, where τu is the subspace topology induced from R with standard topol- ogy. [0, 1) being a normal hyperideal of [0, 1], the quotient hyperring [0, 1]/[0, 1) is a topological hyperring with respect to the quotient topology induced by canonical projection Φ : [0, 1] → [0, 1]/[0, 1). Now, consider X = {0, 1} together with the hyperoperation ⊕ and the binary operation � defined as follows: ⊕ 0 1 0 {0} {1} 1 {1} {0,1} � 0 1 0 0 0 1 0 1 Then, (X,⊕,�,τ′) is a topological hyperring, where τ′ = {φ,{0},X}. If we define ψ : [0, 1]/[0, 1) →{0, 1} by ψ(x + [0, 1)) = [x] = the greatest integer less than or equal to x, then, ψ is a topological isomorphism. Example 2.15. For a hyperring (R, +, ·) and for some positive integer n, the collection Mn(R) of all n × n matrices over R forms a Krasner hyper- ring with respect to the hyperaddition ⊕ and multiplication � defined as, for A = (aij),B = (bij) ∈ Mn(R), A⊕B = {C ∈ Mn(R) : C = (cij),cij ∈ aij +bij} and A � B = (aij · bij). Now, replace (R, +, ·) by the topological hyperring ([0, 1], +, ·,τu) of Example 2.4, where τu is the subspace topology induced from R with standard topology. If we topologize Mn([0, 1]) by identifying it with [0, 1]n 2 , then, Mn([0, 1]) is a topological Krasner hyperring. In a similar man- ner, we can also obtain the topological Krasner hyperring Mn(Mn([0, 1])). Now, let A =   a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44   be an element of M4([0, 1]) and con- sider the map f : M4([0, 1]) → M2(M2([0, 1])) defined by © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 207 M. Singha and K. Das f(A) =   ( a11 a12 a21 a22 ) ( a13 a14 a23 a24 ) ( a31 a32 a41 a42 ) ( a33 a34 a43 a44 )  . Then, f is a topological isomorphism. Theorem 2.16. Let H1 and H2 be two topological hyperrings such that the open subsets of H1 are complete parts. Let f be an open, continuous and good topological homomorphism from H1 onto H2 such that ker f is normal in H1. Then, H1/ ker f and H2 are topologically isomorphic. Proof. Clearly, ψ : H1/ ker f → H2 defined by ψ(x + ker f) = f(x), for all x ∈ H1 is the required isomorphism[24, 7]. So, it is only required to prove that ψ is open and continuous. Consider an open subset U of H1/ ker f. Then, by Remark 2.10 there exists an open subset U of H1 such that U = U/ ker f. So, ψ(U) = ψ(U/ ker f) = f(U), which is open in H2. To prove ψ continuous, consider an open subset V of H2. Then, ψ−1(V ) = {x + ker f : ψ(x + ker f) ∈ V} = {x + ker f : f(x) ∈ V} = f−1(V )/ ker f, which is open in H1/ ker f as f−1(V ) is open in H1. � Theorem 2.17. Let f be a homomorphism from a topological hyperring (H, +, ·,τ) into a topological hyperring (H′, +′, ·′,τ′) such that ker f(6= φ) is normal in H and members of τ are complete parts. Let J be a normal hyperideal of H such that J ⊆ ker f. Then, the homomorphism g from H/J to H′ satisfying g ◦ ΦJ = f is continuous (open, a topological homomorphism) if and only if f is. In particular, if J = ker f, g is a topological isomorphism (good monomor- phism) if and only if f is a topological good epimorphism (good homomorphism). Proof. Continuity and openness are consequences of Theorem 5.3 of [33, p. 33] and Theorem 2.9. To prove ker g 6= φ, take x ∈ ker f. Then, f(x) = 0H′ , which implies g(x + J) = 0H′ . So, x + J ∈ ker g. For the second part, suppose J = ker f. Let f be a topological good homo- morphism. Then, for x,y ∈H, g((x + J) ⊕ (y + J)) = g({z + J : z ∈ x + y}) = {(g ◦ ΦJ )(z) : z ∈ x + y} = f(x + y) = f(x) +′ f(y) = (g ◦ ΦJ )(x) +′ (g ◦ ΦJ )(y) = g(x + J) +′ g(y + J) and g((x + J) � (y + J)) = g((x · y) + J) = (g ◦ ΦJ )(x · y) = f(x · y) = f(x) ·′ f(y) = (g ◦ ΦJ )(x) ·′ (g ◦ ΦJ )(y) = g(x + J) ·′ g(y + J). Now, g(0H + J) = (g ◦ ΦJ )(0H) = f(0H) = 0H′ . Therefore, 0H + J ∈ ker g. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 208 Topological Krasner hyperrings with special emphasis on isomorphism theorems Let x ∈H be such that g(x+J) = 0H′ . Then, f(x) = 0H′ and hence x ∈ ker f, which implies x + ker f = 0H + ker f. Clearly, g is surjective if and only if f is. For the converse, suppose g is a topological good monomorphism. Then, for x,y ∈H, f(x + y) = (g ◦ ΦJ )(x + y) = g(ΦJ (x) ⊕ ΦJ (y)) = g((x + J) ⊕ (y + J)) = g(x + J) +′ g(y + J) = (g ◦ ΦJ )(x) +′ (g ◦ ΦJ )(y) = f(x) +′ f(y). and f(x · y) = (g ◦ ΦJ )(x · y) = g((x · y) + J) = g((x + J) � (y + J)) = g(x + J) ·′ g(y + J) = (g ◦ ΦJ )(x) ·′ (g ◦ ΦJ )(y) = f(x) ·′ f(y). � Corollary 2.18. Let I,J be normal hyperideals of a topological hyperring H such that J ⊆ I and the open subsets of H are complete parts. Then, the following results hold: (1) the map f : H/J →H/I defined by f(x + J) = x + I is a topological good epimorphism. (2) (H/J)/(I/J) and H/I are topologically isomorphic. Proof. (1) Here, f ◦ ΦJ = ΦI is an open, continuous good epimorphism from H → H/I by Theorem 2.9. Then, by Theorem 2.17, f is a topological good epimorphism. (2) As f is surjective, so, ker f 6= φ and ker f = {x + J : f(x + J) = 0 + I} = {x + J : f ◦ ΦJ (x) = I} = {x + J : ΦI(x) = I} = {x + J : x + I = I} = {x + J : x ∈I} = I/J . Then, by Theorem 2.17, g : (H/J)/(I/J) → H/I satisfying g ◦ ΦI/J = f is the required topological isomorphism. � Corollary 2.19. Let I,J be hyperideals of a topological hyperring H such that • J is normal and I is open in H; • open subsets of H are complete parts. Then, I/(I∩J) is topologically isomorphic to (I + J)/J . © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 209 M. Singha and K. Das Proof. Here, f : I → (I + J)/J defined by f(x) = x + J , for all x ∈ I is an epimorphism such that ker f = {x ∈I : f(x) = 0 + J} = {x ∈I : x + J = J} = {x ∈I : x ∈J} = I∩J 6= φ. As I is open in H, by Theorem 22.1 [22, p. 140] f is a quotient map and hence, f is a topological good epimorphism (by Theorem 2.9). So, by Theorem 2.17, g : I/(I∩J) → (I +J)/J satisfying g◦ΦI∩J = f is the required topological isomorphism. � Theorem 2.20. Let K be a complete part dense subhyperring of a topological hyperring H, and let I be a closed normal hyperideal of K, I its closure in H. If the open subsets of H are complete parts, then g(x +I) = x +I is a topological isomorphism from K/I to the dense subhyperring (K + I)/I of H/I. Proof. As the open subsets of H are complete parts, Lemma 3.2 ([11]), continu- ity of multiplication map and Lemma 3.9 ([28]) imply I is a normal hyperideal of H. Here, the kernel of the restriction ΦI|K : K→ (K + I)/I is ker ΦI|K = {x ∈K : ΦI(x) = I} = {x ∈K : x + I = I} = K∩I = I 6= φ and g satisfies g ◦ ΦI = ΦI|K. ΦI|K is a continuous good epimorphism, so, by Theorem 2.17, g is a continuous isomorphism from K/I to (K +I)/I. As K is dense in H and ΦI is continuous, ΦI(K) = (K + I)/I is dense in H/I. Consider an open subset O of K/I and let Φ−1I (O) = P . Then, Remark 2.10 and (1) of Theorem 2.9 imply g(O) = ΦI(P) and P + I = P . P being open in K, there exists an open subset U of H such that P = U ∩K. Claim that (U + I) ∩ K = P . To prove (U + I) ∩ K ⊆ P , let k ∈ K such that k ∈ (U + I). Then, there exist u ∈ U and h ∈ I such that k ∈ u + h. U being a neighborhood of u, there exists a symmetric neighborhood V of zero such that u + V ⊆ U. Then, (h + V ) ∩I 6= φ, so there exists v ∈ V such that (h+v)∩I 6= φ. As I is a closed subset of H, I is also a complete part of H and hence (h+v) ⊆I. Consequently, u+h ⊆ (u−v)+(v+h) ⊆ (u+V )+I ⊆ U +I. As K is a complete part, u + h ⊆ K and hence, by Proposition 2.3 ([28]) u + h ⊆ (U + I) ∩K = (U + I) ∩ (K + I) = (U ∩K) + I = P + I = P . The reverse inclusion also follows from Proposition 2.3 ([28]). Therefore, g(O) = ΦI(P) = ΦI(U)∩((K+I)/I), an open subset of (K+I)/I, for if x+I ∈ ΦI(U) where x ∈ K, then x ∈ Φ−1 I (ΦI(U)) ∩K = (U + I) ∩K = P , which implies x + I ∈ ΦI(P) = g(O). � Conclusion. As in algebra, distinguishing and classifying topological hyper- rings is of great importance in the theory of topological Krasner hyperring too. © AGT, UPV, 2022 Appl. Gen. 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