Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 39, 2020, pp. 147-163 147 Frattini submultigroups of multigroups Joseph Achile Otuwe* Musa Adeku Ibrahim† Abstract In this paper, we introduce and study maximal submultigroups and present some of its algebraic properties. Frattini submultigroups as an extension of Frattini subgroups is considered. A few submultigroups results on the new concepts in connection to normal, characteristic, commutator, abelian and center of a multigroup are established and the ideas of generating sets, fully and non-fully Frattini multigroups are presented with some significant results. Keywords: Maximal, Cyclic Multigroup, Commutator and Generating Set. 2010 AMS subject classification: 55U10.‡ * Department of Mathematics, College of Science (Ahmadu Bello University, Zaria, Nigeria), talk2josephotuwe2016@gmail.com. † Department of Mathematics, College of Science (Ahmadu Bello University, Zaria, Nigeria), amibrahim@abu.edu.ng. ‡ Received on October 13th, 2020. Accepted on December 19th, 2020. Published on December 31st, 2020. doi: 10.23755/rm.v39i0.534. ISSN: 1592-7415. eISSN: 2282- 8214. ©Otuwe et al. et al. This paper is published under the CC-BY licence agreement. J. A Otuwe and M.A Ibrahim. 148 1 Introduction The term multigroup was first mentioned in [7] as an algebraic structure that satisfied all the axioms of group except that the binary operation is multivalued. This concept was later redefined in [19] via count function of multisets and some of its properties were vividly discussed. The idea of submultigroup and its classes were established in [14]. Concept of maximal subgroups is established in [3], [5] and [17] and some of its properties were investigated. Also, normal and characteristic submultigroup were introduced in [8] and [13] respectively, and some of its properties were presented. Frattini in [11], introduced a special subgroup named Frattini subgroup and some results were obtained. Other related work on Frattini subgroup can be found in [1], [2], [6], [12], [15], [16], [18], [20] and [21]. Furthermore, in [10] Frattini subgroup was represented but in fuzzy environment called Frattini fuzzy subgroup. In this paper, we focus on multiset setting to obtain Frattini submultigroups and finally establish some related results. In general, the union of submultigroups of a multigroup may not be a multigroup, we therefore establish some conditions under which the union of all maximal submultigroups is a multigroup. When this occur, the Frattini submultigroup obtained from such maximal submultigroups is called “fully Frattini” otherwise it is called “non-fully Frattini”. Furthermore, other relevant concepts such as; cyclic multigroup, minimal generating set of a multigroup, generator and non-generator of a multigroup are introduced with reference to Frattini submultigroups. Finally, we study some properties of center of a multigroup, normal, commutator, minimal and characteristic submultigroups. 2 Preliminaries Definition 2.1 (|23|). Let be a set. A multiset over is just a pair , where is a set and is a function. Any ordinary set is actually a multiset , where is its characteristic function. The set is called the ground or generic set of the class of all multisets containing objects from . Definition 2.2 (|22|). Let and be two multisets over , is called a submultiset of written as if for all . Also, if and , then is called a proper submultiset of and denoted as . Frattini submultigroups of multigroups 149 Definition 2.3 (|22|). Let and be two multisets over , then and are equal if and only if for all . Two multisets and are comparable to each other if or . Definition 2.4 (|23|). Suppose that , such that and . i. Their intersection denoted by is the multiset , where , . ii. Their union denoted by is the multiset , where , . iii. Their sum denoted by is the multiset , where , . Definition 2.5 (|19|). Let be a group and . is said to be a multigroup of 𝑋 if the count function of or satisfies the following two conditions: i. (𝑥𝑦) ≥ (𝑥) (𝑦)], ∀𝑥, 𝑦 ∈𝑋. ii. ( ) ≥ (𝑥), ∀ 𝑥 ∈𝑋, Where is a function that takes to a natural number, and denotes minimum operation. The set of all multigroups defined over 𝑋 is denoted by (𝑋). Definition 2.6 (|19|). Let . Then is defined by . Thus, . Definition 2.7 (|19|). Let . Then is said to be abelian or commutative if . Definition 2.8 (|19|). Let . Then the sets and are defined as and , where is the identity element of . Definition 2.9 (|19|). Let , be an arbitrary family of multigroups of a group Then J. A Otuwe and M.A Ibrahim. 150 Definition 2.10 (|14|). Let . Then the center of is defined as . Definition 2.11 (|9|) Commutator of two Submultigroup: Let and be submultigroups of . Then the commutator of and is the multiset of defined as follows: That is, . Since the supremum of an empty set is zero. if is not a commutator. Definition 2.11 (|4|). Let . Then the order of denoted by is defined as . i.e., the total numbers of all multiplicities of its element. Definition 2.12 (|14|). Let . A submultiset of is called a submultigroup of denoted by if is a multigroup. A submultigroup of is a proper submultigroup denoted by , if and Definition 2.13 (|14|). Let . Then a submultigroup of is said to be complete if , incomplete if , regular complete if is complete and and regular incomplete if is incomplete and . Definition 2.14 (|8|). Let such that . Then is called a normal submultigroup of if . Definition 2.15 (|10|). Let and be two groups and let be a homomorphism. Suppose and are multigroups of and respectively, then induces a homomorphism from to which satisfies i. . Frattini submultigroups of multigroups 151 ii. where i. the image of under denoted by , is a multiset of defined by for each ii. the inverse image of under denoted by , is a multiset of defined by . Definition 2.16 (|10|). Let and be groups and let and respectively. Then a homomorphism from to is called an automorphism of onto if is both injective and surjective, that is, bijective. Definition 2.17 (|13|). Let such that . Then is called a characteristic (fully invariant) submultigroup of if for every automorphism, of . That is, for every . 3 Frattini Submultigroups and their Properties In this section we propose the concept of minimal, maximal, Frattini, commutator submultigroups, cyclic, fully and non-fully Frattini multigroup and generating set of a multigroup with some illustrative examples. Definition 3.1 a. Minimal Submultigroup: Let be a group and . Then a non trivial proper submultigroup denoted by of is said to be minimal if there exists no other non-trivial submultigroup of such that . Remark 3.1. Every minimal complete submultigroup of a multigroup is unique. b. Maximal Submultigroup: Let be a group and . Then a proper normal submultigroup denoted by of is said to be maximal if there exists no other proper submultigroup of such that . J. A Otuwe and M.A Ibrahim. 152 c. Frattini Submultigroup: Let be a group. Suppose is a multigroup of and, , , , (or simply for ) are maximal submultigroups of . Then the Frattini submultigroup of denoted by is the intersection of defined by or simply by . Remark 3.2. i. Let be a non abelian group and . If is a normal submultigroup of with an incomplete maximal submultigroups and for each are the maximal subgroups of , then the maximal submultigroups of are submultigroups of . ii. Let . If is a submultigroup of and is the Frattini submultigroup of then, . d. Commutator Submultigroup of a Multigroup: Let such that the commutator subgroup of is given as . Then the commutator submultigroup of denoted by is defined as e. Let . Then the sets and are defined as and , where is the identity element of . Remark 3.3. i. The commutator submultigroup of every abelian multigroup is ii. Let and be the commutator submultigroup of . Then . Remark 3.4. Let such that and be the commutator submultigroup of and . Then . f. Cyclic Multigroup: Let be a group generated by . Then a multigroup over is said to be a cyclic multigroup if such that . The element is then called the generator of otherwise, a non generator of . Frattini submultigroups of multigroups 153 g. Generating Set of a Multigroup: Let be a cyclic group and . A subset of is said to be a generating set for if all elements of and its inverses can be expressed as a finite product of elements in with and for some . h. Minimal Generating Set of a Multigroup: Let be a cyclic group and . A subset of is termed minimal generating set of if such that and there is no proper subset of with , . i. Fully Frattini Multigroup: Let . Then is called fully Frattini if the union of the maximal submultigroups equals . Otherwise, it is called non-fully Frattini. In addition, every multigroup without incomplete maximal submultigroup is called trivial fully Frattini. 4. Some Results on Frattini Submultigroups In this section, we present some results on Frattini submultigroup of multigroups. Theorem 4.1. Let with complete maximal submultigroups. Then every minimal submultigroup of is a submultigroup of . Proof. Suppose is the Frattini submultigroup of then has maximal submultigroup say such that . Since is multigroup over , has a minimal submultigroup say . If is not a submultigroup of then there exist at least an element such that which contradicts the fact that is a minimal submultigroup of . Hence is a submultigroup of Theorem 4.2 If for all , then is characteristic in . Proof. Since is an automorphism, the inverse is also an automorphism of . Hence we have . Applying , we have . Then we obtain . By this J. A Otuwe and M.A Ibrahim. 154 fact, equality holds and so . Hence the Frattini submultigroup is characteristic in . Theorem 4.3 Every Frattini submultigroup of a multigroup is characteristic. Proof. By Theorem 4.2, it suffices to proof that for every automorphism . Let . Then there exists such that . To show that , we consider an arbitrary element . Then since is an automorphism, we have . Thus there exists in such that . We have (Since is a homomorphism) (Since ) (Since is a homomorphism) Since this is true for all it follows that , and thus . Hence the result. Theorem 4.4 Every Frattini submultigroup of a multigroup is abelian. Proof. Let and be the Frattini submultigroup of . It follows that is a normal submultigroup of by definition 2.14 Consequently, . Thus, . Hence, the result follows by Definition 2.7 Theorem 4.5 Every is a normal submultigroup of . Proof. Let and be the Frattini submultigroup of . Then Frattini submultigroups of multigroups 155 , since . Now, let , then since is a multigroup over by definition 2.14, we get Now we proof that is a normal submultigroup of . Let , then it follows that . Hence, the result by Definition 2.14 Theorem 4.6 Let be a multigroup over a non-Abelian group , then . Proof. , since at least . Let then for all , and . Consequently, where since . Thus . Now, let Then . Hence, . Thus, therefore, is a subgroup of . To show that is a normal subgroup of . Let and Then . J. A Otuwe and M.A Ibrahim. 156 Thus, and . Hence, Remark 4.1 If is a multigroup over an abelian group , is the root set of Frattini submultigroup of and is the center of then is a normal subgroup of . Theorem 4.7. If is a multigroup over a non-Abelian group and is a normal submultigroup of , then . Proof. Clearly, and are submultigroups of . Let be the maximal submultigroups of and be the maximal submultigroups of for each and . Then by Remark 3.2 ( ) we have for each and . | Therefore, . To show that , suppose then the result holds trivially. But if then for any element , for each and . Therefore, . Theorem 4.8 If is a regular multigroup over a group . Then . Proof. Since is a multigroup over , then . Let and then . Thus . Therefore is a submultigroup of . Now by Theorem 4.5, and clearly . So let and , then implies . Hence . Frattini submultigroups of multigroups 157 Theorem 4.9 If , is the commutator submultigroup of and is the Frattini submultigroup of . Then . Proof. Since is a multigroup over , then . Let and , then . Thus . Therefore is a submultigroup of . Now, by Theorem 4.5, . Let and , then implies . Hence, . Theorem 4.10. Every Frattini submultigroup of a cyclic multigroup is abelian. Proof. Let be the Frattini submultigroup of a cyclic multigroup over a cyclic group , then there exists such that we have and for . It now follows that . Theorem 4.11. If is a regular multigroup with an incomplete maximal submultigroups over a cyclic group . Then is contained in the set of all non-generators of . In particular, coincide with the set of all non- generators if has only one maximal submultigroup. Proof. Let be a cyclic group and and denotes the Frattini submultigroup of . Let be the set of all generators of and be the incomplete maximal submultigroups of , then for all , In fact, all is a non-generator. Further, and . Since , we have that J. A Otuwe and M.A Ibrahim. 158 for all , . This implies that . But is the largest set containing all non generators. Hence is contained in the set of all non-generators. Suppose has only one nontrivial maximal submultigroup say then, and . Since , therefore for all non-generators . Hence, is indeed the set of all non-generators. Theorem 4.12. If a regular multigroup over a cyclic group has two incomplete maximal submultigroups, then the union of its generators coincide with the non-generating set of . Proof. Let and be the maximal submultigroups of and be the collection of all generators of . Clearly, and (since and does not contain any generator). Now, can be expressed as if is odd and if is even with for any maximal submultigroup of . Also, . That is, for odd values of and for any but since for even we have for any . Hence the result. Theorem 4.13. If a regular multigroup over a cyclic group has two maximal submultigroups, then the union of the non-generators coincide with the generating set of . Proof. Let and be the maximal submultigroups of and be the collection of all non- generators of . Clearly, and so generates . , where . Frattini submultigroups of multigroups 159 Taking every , (for some ) if (i.e., generates distinct elements in ). Since we have that for some . More explicitly, where . This yields for some . Theorem 4.14. If a regular multigroup over a cyclic group has two incomplete maximal submultigroups and is the set of generators of , then form one of the root set of the maximal submultigroup of for some . Proof. Let be a multigroup over a cyclic group, be the maximal submultigroups of and be the generators of . Then, . Since , for all , , and But, . Therefore, . In particular, and for any . Remark 4.2 a. A generator of any multigroup over a cyclic group is not contained in any of its maximal submultigroups. b. The set of non-generators of any multigroup may not be a submultigroup. Theorem If is a minimal generating set of a multigroup over a cyclic group , then . J. A Otuwe and M.A Ibrahim. 160 Proof. Suppose , then , where is a maximal submultigroup of . Now, since contains at least one generator of , then every contains at least one generator of which is a contradiction. Hence, . Remarks 4.3 i. If is a multigroup over a cyclic group . Then the union of all the minimal generating sets of is equal to . ii. Every minimal generating set contains a non-generator. iii. Given a multigroup over a cyclic group with order , If is a minimal generating set of then gives elements of . Theorem 4.16 Every irregular multigroup with complete maximal submultigroups over a group is fully Frattini. Proof. For multigroup to be irregular implies , . Now let for each be the complete maximal submultigroups of . For to be complete in implies . Since is complete in , then there exists such that . Therefore for each . Theorem 4.17 Every irregular multigroup with an incomplete maximal submultigroups over a non-cyclic group is fully Frattini. Proof. is a non-cyclic group, implies it has no generator and . Now let for each be the incomplete maximal submultigroups of . Then for each , is contained in at least one of the with Theorem Every cyclic multigroup with incomplete maximal submultigroups is not fully Frattini. Proof. Suppose is a cyclic group and is a multigroup with incomplete maximal submultigroups over . Where has set of generators . Now Frattini submultigroups of multigroups 161 let for some finite then , where are the root sets of all the maximal submultigroups of for each and .by remark 4.2 , Hence, is not fully Frattini. Theorem 4.19 Every regular multigroup over a group is non-fully Frattini. Proof. Let be a group and be a multigroup over . For to be a regular multigroup implies , . Let for each be the maximal submultigroup of . Since is regular and by Definition , there exists at least an element such that for each and so Remark 4.4. Let be a nontrivial fully Frattini multigroup over a group then has at least three maximalsubmultigroups if where is the identity element of . 5 Conclusions Most results in Frattini subgroup are extended to multigroup. A number of new results were obtained. 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