A study of a generalization of bi-ideal , quasi ideal and interior ideal of semigroup

In this paper, as a further generalization of ideals, we introduce the notion of biquasi-interior ideals as a generalization of biideal, quasi ideal, interior ideal, bi-interior ideal and bi-quasi ideal of semigroup and study the properties of bi-quasi-interior ideals of semigroup.


Introduction
Ideals play an important role in advance studies and uses of algebraic structures.Generalization of ideals in algebraic structures is necessary for further study of algebraic structures.Many mathematicians proved important results and characterizations of algebraic structures by using the concept and the properties of generalization of ideals in algebraic structures.Henriksen [2] and Shabir et al. [15] studied ideals in semirings.
We know that the notion of an one sided ideal of any algebraic structure is a generalization of notion of an ideal.The quasi ideals are generalization of left ideal and right ideal whereas the bi-ideals are generalization of quasi ideals.
In 1952, the concept of bi-ideals was introduced by Good and Hughes [1] for semigroups.The notion of bi-ideals in rings and semigroups were introduced by Lajos and Szasz [8,9].Bi-ideal is a special case of (m − n) ideal.In 1956, Steinfeld [16] first introduced the notion of quasi ideals for semigroups and then for rings.Iseki [5,6,7,8] introduced the concept of quasi ideal for a semiring.Quasi ideals bi-ideals in Γ-semirings studied by Jagtap and Pawar [7].Murali Krishna Rao [10,11,12] introduced the notion of left (right) bi-quasi ideal of semigroup, Γ-semigroup, Γ-semiring and study the properties of left bi-quasi ideals.
Murali Krishna Rao [13,14] studied ideals in ordered Γ-semirings and introduced the notion of bi-interior ideal as a generalization of quasi ideal, bi-ideal and interior ideal of semigroup and study the properties of bi-interior ideals of semigroup, simple semigroup and regular semigroup.
In this paper, as a further generalization of ideals, we introduce the notion of bi-quasi-interior ideal as a generalization of bi-ideal,quasi ideal, interior ideal, bi-interior ideal and bi-quasi ideal of semigroup and study the properties of bi-quasi-interior ideals of semigroup and some characterizations of bi-quasi-interior ideals of semigroup, regular semigroup and simple semigroup.
A necessary and sufficient condition for a semigroup to be regular and simple is proved using bi-quasi-interior ideals semigroup.

Preliminaries
In this section we will recall some of the fundamental concepts and definitions, which are necessary for this paper.Definition 2.1.Let M be a semigroup.If there exists 1 ∈ M such that a • 1 = 1 • a = a, for all a ∈ M, is called an unity element of M then M is said to be semigroup with unity.Definition 2.2.An element a of a semigroup M is called a regular element if there exists an element b of M such that a = aba.Definition 2.3.A semigroup M is called a regular semigroup if every element of M is a regular element.Definition 2.4.An element a of a semigroup M is called an idempotent element if aa = a.
Definition 2.10.Let M be a semigroup.L is said to be bi-quasi ideal of M if it is both a left bi-quasi and a right bi-quasi ideal of M.
Example 2.1.(i) Let Q be the set of all rational numbers.Then and Γ = M then M is a semigroup with respect to usual matrix multiplication and Then A is not a bi-ideal of semigroupM.

Bi-quasi-interior ideals of semigroups
In this section we introduce the notion of bi-quasi-interior ideal as a generalization of bi-ideal,quasi-ideal and interior ideal of semigroup and study the properties of bi-quasi-interior ideal of semigroup.
Every bi-quasi-interior ideal of semigroupM need not be bi-ideal,quasiideal, interior ideal bi-interior ideal.and bi-quasi ideals of semigroup M.
Example 3.1.Let N be a set of all natural numbers.N is a semigroup with respect to usual addition of integers.A subset I = 2N of N is a bi-quasiinterior ideal of N but not bi-ideal,quasi-ideal, interior ideal, bi-interior ideal and bi-quasi ideal of semigroup N .Theorem 3.1.Every bi-ideal of semigroup M is a bi-quasi-interior ideal of semigroupM.
Proof.Let B be a bi-ideal of semigroupM.Then BM B ⊆ B.
Therefore BM BM B ⊆ BM B ⊆ B.
Hence every bi-ideal of semigroup M is a bi-quasi-interior ideal of M.
Theorem 3.2.Every interior ideal of semigroupM is a bi-quasi-interior ideal of M.
Proof.Let I be an interior ideal of semigroupM.
Then IM IM I ⊆ M IM ⊆ I.
Hence I is a bi-quasi-interior ideal of semigroup M.
Theorem 3.3.Let M be a semigroup.Every left ideal is a bi-quasi-interior of M.
Proof.Let B be a left ideal of semigroupM.Then M B ⊆ B.
Hence every left ideal of semigroup M is a bi-quasi-interior ideal of M.
Corollary 3.1.Let M be a semigroup.Every right ideal is a bi-quasiinterior of M.
Corollary 3.2.Let M be a semigroup.Every ideal is a bi-quasi-interior ideal of M.
In the following theorems, we mention some important properties and we omit the proofs since proofs are straight forward.Theorem 3.4.Let M be a semigroup.Every quasi ideal is bi-quasi-interior of M. Theorem 3.5.Let M be a semigroup.Intersection of a right ideal and a left ideal of M is a bi-quasi-interior ideal of M. Theorem 3.6.Let M be a semigroup.If L is a left ideal and R is a right ideal of semigroup M and B = RL then B is a bi-quasi-interior ideal of M. Theorem 3.7.Let M be a semigroup.If B is a bi-quasi-interior ideal and T is a non-empty subset of M such that BT is a subsemigroup of M then BT is a bi-quasi-interior ideal of semigroup M. Conversely suppose that there exist L and R are left and right ideals of Hence B is a bi-quasi-interior ideal of semigroup M.
Theorem 3.11.The intersection of a bi-quasi-interior ideal B of semigroup M and a right ideal A of M is always bi-quasi-interior ideal of M.
Hence the intersection of a bi-quasi-interior ideal B of semigroup M and a subsemigroup A of M is always a bi-quasi-interior ideal of M. Hence BT is a bi-quasi-interior ideal of semigroup M.
Theorem 3.17.Let B be bi-ideal of semigroup M and I be an interior ideal of M. Then B ∩ I is a bi-quasi-interior ideal of M.
Proof.Obviously B ∩ I is subsemigroup of M. Suppose B is a bi-ideal of M and I is an interior ideal of M. Then

Theorem 3 . 8 .
Let M be a semigroup and B be a subsemigroup of M .If BM M M B ⊆ B then B is a bi-quasi-interior ideal of M. Theorem 3.9.Let M be a semigroup and B be a subsemigroup ofM .If M M M M B ∩ BM M M M ⊆ B then B is a bi-quasi-interior ideal of M.Theorem 3.10.Let M be a semigroup and B be a subsemigroup of M .B is a bi-quasi-interior ideal of M if and only if there exist a left ideal L and a right idealR of M such that RL ⊆ B ⊆ R ∩ L .Proof.Suppose B is a bi-quasi-interior ideal of semigroupM .Then BM BM B ⊆ B. Let R = BM BM and L = M B.Then L and R are left and right ideals of M respectively.Therefore RL ⊆ B ⊆ R ∩ L .

Theorem 3 . 12 .
Let A and C be bi-quasi-interior ideals of semigroup M and B = AC.If CC = C then B is a bi-quasi-interior ideal of M. Proof.Let A and C be bi-quasi-interior ideals of semigroup M and B = AC BB = ACAC = ACCCAC ⊆ ACM CM C ⊆ AC = B. Therefore B = AC is a subsemigroup of M BM BM B = ACM ACM AC ⊆ AM AM AC ⊆ AC = B. Hence B is a bi-quasi-interior ideal of M. Corollary 3.3.Let A and C be bi-quasi-interior ideals of semigroup M and B = CA.If CC = C then B is a bi-quasi-interior ideal of M. Theorem 3.13.Let A and C be subsemigroups of M and B = AC.If A is the left ideal then B is a bi-quasi-interior ideal of M. Proof.Let A and C be subsemigroups of M and B = AC Suppose A is the left ideal of M. BB = ACAC ⊆ AC = B. BM BM B = ACM ACM AC ⊆ AC = B. Hence B is a bi-quasi-interior ideal of M. Corollary 3.4.Let A and C be subsemigroups of semigroup M and B = AC.If C is a right ideal then B is a bi-quasi-interior ideal of M. Theorem 3.14.Let M be a semigroup and T be a non-empty subset of M. Then every subsemigroup of T containing T M T M T is a bi-quasi-interior ideal of semigroup M. Proof.Let B be a subsemigroup of T containing T M T M T. Then BM BM B ⊆ T M T M T ⊆ B. Therefore BM BM B ⊆ B. Hence B is a bi-quasi-interior ideal of M. Theorem 3.15.B is a bi-quasi-interior ideal of semigroup M if and only if B is a left ideal of some right ideal of semigroupM .Proof.Suppose B is a left ideal of some right ideal R of semigroupM .Then RB ⊆ B, RM ⊆ R. Hence BM BM B ⊆ BM B ⊆ RM B ⊆ RB ⊆ B. Therefore B is a bi-quasi-interior ideal of semigroup M .Conversely suppose that B is a bi-quasi-interior ideal of semigroup M .Then BM BM B ⊆ B. Therefore B is a left ideal of right ideal BM BM of semigroupM .Corollary 3.5.B is a bi-quasi-interior ideal of semigroup M if and only if B is a right ideal of some left ideal of semigroupM .Theorem 3.16.If B is a bi-quasi-interior ideal of semigroup M, T is a subsemigroup of M and T ⊆ B then BT is a bi-quasi-interior ideal of M. Proof.Suppose B is a bi-quasi-interior ideal of semigroup M, T is a subsemigroup of M and T ⊆ B. BT BT ⊆ BT.Hence BT is a subsemigroup of M. We have M BT M ⊆ M BM and BT M BT ⊆ BM B, then BT M BT M BT ⊆ BM BM BT ⊆ BT .

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B ∩ I)M (B ∩ I)M (B ∩ I) ⊆ BM BM B ⊆ B, (B ∩ I)M (B ∩ I)M (B ∩ I) ⊆ IM IM I ⊆ I Therefore (B ∩ I)M (B ∩ I)M (BI)M ⊆ B ∩ I.Hence B ∩ I is a bi-quasi-interior ideal of M.Theorem 3.18.Let M be a semigroup and T be a subsemigroup of M. Then every subsemigroup of T containing T M T M T is a bi-quasi-interior ideal of M.Proof.Let C be a subsemigroup of T containing T M T M T. ThenCM CM C ⊆ T M T M T ⊆ C.Hence C is a bi-quasi-interior ideal of semigroup M .Theorem 3.19.Let M be a semigroup.If M = M a, for all a ∈ M , then every bi-quasi-interior ideal of M is a quasi ideal of M.