As part of his study of representations of the polycylic monoids, M.V. Lawson described all the closed inverse submonoids of a polycyclic monoid Pn and classified them up to conjugacy. We show that Lawson's description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply B. Schein's theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.
Let a, b, x, y, m, n be positive integers. Many special cases of the diophantine equation axwhere (ax, by) = 1, a, b are square-free integers, y > 1, m is odd and n is an odd prime, have been considered in the last few years (see [1, 3,[7][8][9][10][11]). Le Maohua [4-6] studied this equation in full generality and proved that it has only a finite number of solutions (a, b, x, y, m, n) with n > 5. Following almost the same method as Le Maohua but using a recent result of Bilu, Hanrot and Voutier [2], we are able to prove:Theorem. The diophantine equationwhere a, b, x, y, k, n are positive integers such that (ax, b) = 1, a, b are squarefree integers, k ≥ 0, n is an odd prime, (n, h) = 1 where h is the class number of the field Q( √ −ab) and y > 1, has no solutions in (a, b, x, y, k, n) when n > 13 and has exactly six solutions for 7 ≤ n ≤ 13, given by (a, b, k, n, y) = (1, 7, 0, 13, 2), (1,7, 1,7, 2), (1, 19, 0,7, 5), (3, 5, 0, 7, 2), (5, 7, 1, 7, 3), (13, 3, 0,7, 4).Further if a = 1, n = 5, then (1) has exactly 2 solutions given by k = 0 and (b, y) = (7, 2), (11, 3).We are grateful to Professor Bilu for his valuable suggestions and also for providing us with a copy of [1]. He also informed us that similar results using similar approach have been obtained by Bugeaud [3]. Our paper was submitted independently although later than that of Bugeaud.We start by giving some important definitions.
Let S be an inverse semigroup with semilattice of idempotents E(S). Recall that the natural partial order on S is defined by s t ⇐⇒ there exists e ∈ E(S) such that s = et .A subset A ⊆ S is closed if, whenever a ∈ A and a s, then s ∈ A. The closure B ↑ of a subset B ⊆ S is defined asAn atlas in S is a subset A ⊆ S such that AA −1 A ⊆ A: that is, A is closed under the heap ternary operation a, b, c = ab −1 c (see [3]). Since, for all a ∈ A we have a, a, a = a, we see that A is an atlas if and only ifProposition 2.2. [18, Proposition 5.] A coset C that contains an idempotent e ∈ E(S) is an inverse subsemigroup of S, and in this case C = (CC −1 ) ↑ . Proof. If a, b ∈ C then ab aeb = a, e, b ∈ C and since C is closed, we have ab ∈ C. Furthermore, a −1 ea −1 e = e, a, e ∈ C and so a −1 ∈ C. Hence C is an inverse subsemigroup. Now ab −1 ∈ CC −1 and ab −1 ab −1 e = a, b, e ∈ C. Since C is closed we have (CC −1 ) ↑ ⊆ C. But if x ∈ C then x xe ∈ CC −1 and so x ∈ (CC −1 ) ↑ . Therefore C = (CC −1 ) ↑ .
Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy G-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy G-semiprimary ideal of a ring R under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under G-homomorphism of fuzzy ideals.
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