Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

Introduction

Given any invertible matrices A and B, it is well-known that (AB)⁻¹ = B⁻¹A⁻¹. This property is often called reverse order law for the invertible matrices. This law is easily extended for the invertible elements from an associative ring with identity. This notion has been well studied in the literature and it is known that the analogue is not true in the case of generalized inverses, unless there are certain additional conditions. Several researchers came up with many necessary as well as sufficient conditions for the reverse order law to hold in the case of different generalized inverses. Ivan Erdelyi in 1966 discussed reverse order law for Moore-Penrose inverses of complex matrices in the context of partial isometries¹. Greville, in his celebrated paper², gave several equivalent conditions for the reverse order law for Moore-Penrose inverse of matrices over complex field. He proved that given two complex matrices A and B with AB defined, the reverse order law (AB)^† = B^†A^† holds if and only if the equations A^†ABB*A* = BB*A* and BB^†A*AB = A*AB are satisfied, which is equivalent to the column space inclusions 𝒞 (BB*A*) ⊆ 𝒞 (A*) and 𝒞 (A*AB) ⊆ 𝒞 (B). Here, the notations ^† and * denote the Moore-Penrose inverse and conjugate transpose, respectively. Motivated by this paper, Hartwig discussed equivalent conditions for triple reverse order law for Moore-Penrose inverses (see, 3) and Y. Tian discussed equivalent conditions for multiple reverse order law for Moore-Penrose inverses of complex matrices (see, 4).

The reverse order law for Moore-Penrose inverses in the context of elements from an associative ring with involution was studied previously⁵. Also, it has been extended to different generalized inverses including group inverse, Drazin inverse, core inverse, Drazin Moore-Penrose (DMP) inverse and pseudo core inverse, of matrices over a field, operators on Hilbert space, and elements of rings. Interested readers may refer to 4, 6–10 and the references therein.

Another notion of our concern is the star order, introduced by M. P. Drazin in 1978¹¹. In the beginning of 90’s, J. K. Baksalary and S. K. Mitra considered weaker conditions than those considered by Drazin and introduced left-star and right-star orders¹². Benitez et al.,¹³ discussed some equivalent conditions for reverse order law for group inverse and Moore-Penrose inverse in terms of sharp and star partial orders, respectively, where most of his results assume that one of the matrices under consideration is Equal Projector (EP).

In the present paper, one of our main focuses is to study the reverse order law for Moore-Penrose inverses of arbitrary elements, which need not be EP, in terms of different partial orders. While exploring the extension of reverse order law for the class of outer inverses, it may be noted that an outer inverse of given element is not uniquely established, unlike in the cases of inverse, Moore-Penrose inverse and group inverses. So, a natural question arises is that whether every outer inverse (ab)⁼ of ab could be written as g_bg_a for some outer inverses g_b of b and g_a of a. In case the answer is affirmative, then verify if

holds. While answering these questions, a few noteworthy properties of right-star, left-star and star partial orders are studied. Further, we give some equivalent conditions for the reverse order law for Moore-Penrose inverse in terms of different star partial orders.

Preliminaries

Throughout our discussion, 𝒜 denotes an associative ring, need not be with unity unless indicated otherwise. We use the notations a, b, c, ... for the elements of a ring and A, B, C, ... for matrices under discussion. Let a ∈ 𝒜. If there exists a g ∈ 𝒜 satisfying aga = a, then a is said to be regular and g is called a generalized inverse (or simply a g-inverse) of a. An arbitrary g-inverse of a is denoted by a⁻. If there exists a g ∈ 𝒜 satisfying gag = g, then such a g is called an outer inverse of a. An arbitrary outer inverse of a is denoted by a⁼. A generalized inverse, which is also an outer inverse, is called a reflexive generalized inverse and an arbitrary reflexive generalized inverse of a is denoted by $a_r^{-}$. A commuting reflexive generalized inverse of a is called the group inverse of a and it is denoted by a^#. It is unique whenever it exists. The classes {a⁻}, {a⁼}, and {$a_r^{-}$} denote the class of all g-inverses, outer inverses and reflexive g-inverses of a, respectively. For basic notions of generalized inverses in the context of matrices we refer the readers to 14, 15, and 16.

An involution ‘*’ of a ring 𝒜 is an anti-automorphism whose square is identity; i.e.

Involution ‘*’ is said to be proper (see 11) if for all a, b ∈ 𝒜,

A proper *-ring is a ring equipped with a proper involution ‘*’. A ring is said to be regular ring if all of its elements are regular. A *-regular ring is a regular ring with involution ‘*’ such that a*a = 0 ⇒ a = 0. Observe that the involution is proper if and only if a*a = 0 ⇒ a = 0 holds. An important consequence of this property of involution is the *-cancellation law.

An element a is said to satisfy left (right) *-cancellation law if a*ax = a*ay ⇒ ax = ay (xaa* = yaa* ⇒ xa = ya). Element a is said to be *-cancellable if it satisfies both left and right *-cancellation laws.

For an associative ring 𝒜 with involution '*' and a ∈ 𝒜, consider the following four equations:

An element g satisfying the equations {i, j, ...} ⊆ {1, 2, 3, 4}, is called {i, j, ...}-inverse of a and the class of all {i, j, ...}-inverses of a is denoted by {a^{{i, j,...}}}. An element g ∈ 𝒜 satisfying all of (1), (2), (3) and (4), if exists, is unique and is called the Moore-Penrose inverse of a. The existence of Moore-Penrose inverse for the elements of a *-regular ring was proved by Kaplansky¹⁷, almost at the same time that Penrose¹⁸ proved the existence in the case of matrices. The Moore-Penrose inverse of an element a is denoted by a^†. We will use the notation 𝒜^† to denote the set of all elements of 𝒜 having Moore-Penrose inverse. Note that the elements of 𝒜^† are *-cancellable.

An element a is Hermitian if a* = a, and it is an idempotent if a² = a. Hermitian idempotents are called projectors. If a^† is the Moore-Penrose inverse of a, then both aa^† and a^†a are projectors.

The right annihilator a^o of an element a ∈ 𝒜 is the set {x ∈ 𝒜 : ax = 0} and the left annihilator ^oa is the set {y ∈ 𝒜 : ya = 0}. If 1 ∉ 𝒜, then the right annihilator (1 − a)^o is defined as {x ∈ 𝒜 : x = ax} and the left annihilator ^o(1 − a) = {y ∈ 𝒜 : y = ya} (see 19). For any a, b ∈ 𝒜, we write $a\underset{cs}{\simeq }b$ ($a\underset{rs}{\simeq }b$) if the principal right (left) ideals generated by a and b coincide. We say that a and b are space equivalent if both $a\underset{cs}{\simeq }b$ and $a\underset{rs}{\simeq }b$ are true. In such a case we write $a\underset{sp}{\simeq }b$.

The sum a + b is written as a ⊕ b (⊕ refers to direct sum) if a𝒜 ∩ b𝒜 = {0} and 𝒜a ∩ 𝒜b = {0}. If so is the case, then we say that a and b are disjoint. We refer to 19, 20, and 21 for more details on the above notions.

The minus partial order is helpful in discussing many of the properties of star partial order. For the initial works on minus partial order, one may refer to 21–23.

Definition 1 (Minus Partial Order^21,22). The minus partial order is a relation, denoted by ≤⁻, defined on set of all regular elements of 𝒜 by a ≤⁻ b if there exists a g-inverse g of a such that ag = bg and ga = gb.

For the proof of the following theorem, see 24 and 25.

Theorem 2 (Right-left Symmetry^24,25). Let 𝒜 be an associative ring with unity and let a, b, c ∈ 𝒜 such that a = b + c is regular. Then the following statements are equivalent.

Note that the theorem holds even in the case of an associative ring without unity, provided a, b, c are regular.

Definition 3 (Star Order¹¹). Let 𝒜 be an associative ring with involution ‘*’ and a, b ∈ 𝒜. The relation ≤* defined by a ≤* b if a*a = a*b and aa* = ba* is called star order on 𝒜. If a ≤* b, then we say that the element a is dominated by b under star order.

Drazin, in 11 noticed that the relation star order defined on a *-semigroup is a partial order. He pointed out that if the Moore-Penrose inverses of a and b exist, then,

and

He also noted that

Definition 4. A partial order ≤₂ on a set S₂ is said to be dominated by a partial order ≤₁ defined on set S₁ if S₂ ⊆ S₁ and for a, b ∈ S₂,

Note that the minus partial order dominates the star partial order (as seen from (5)).

Reverse order law for outer inverses

In this section we probe if the reverse order law mentioned in the earlier section holds. In the following lemma, we observe that the inclusion {(ab)⁼} ⊆ {b⁼a⁼} always holds.

Lemma 5. For a, b ∈ 𝒜, we have

Proof. Given an outer inverse g_ab of ab, note that g_aba is an outer inverse of b and bg_ab is an outer inverse of a. Further, we have g_ab = g_ab(ab)g_ab = (g_aba)(bg_ab) ∈ {b⁼a⁼}, proving (8). Hence the lemma.

Unfortunately, the reverse of inclusion appearing in (8) need not be true. We have a counter example in the matrix case. For $A=\left[\begin{array}{l}\begin{array}{l}1\\ 2\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}3\\ 6\end{array}\hfill \end{array}\right]$ and $B=\left[\begin{array}{l}\begin{array}{l}2\\ 2\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}1\\ 1\end{array}\hfill \end{array}\right]$, consider $G_A=\left[\begin{array}{l}\begin{array}{l}1\\ 0\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}0\\ 0\end{array}\hfill \end{array}\right]$ and $G_B=\left[\begin{array}{l}\begin{array}{l}0\\ 1\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}0\\ 0\end{array}\hfill \end{array}\right]$, which are outer inverses of A and B, respectively. Note that $G_B{G}_A=\left[\begin{array}{l}\begin{array}{l}0\\ 1\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}0\\ 0\end{array}\hfill \end{array}\right]$ is not an outer inverse of $AB=\left[\begin{array}{l}\begin{array}{l}8\\ 16\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}4\\ 8\end{array}\hfill \end{array}\right]$, as we have

This proves that {B⁼A⁼} ≠ {(AB)⁼}.

In 26 and 27, the authors presented necessary and sufficient conditions for {B^{1,2,3}A^{1,2,3}} ⊆ {(AB)^{1,2,3}} in the case of matrices and bounded linear operators on Hilbert spaces, respectively. An equivalent condition for the inclusion {B^{1,2}A^{1,2}} ⊆ {(AB)^{1,2}} is given in 28 for bounded linear operators on complex Hilbert spaces.

Now, our interest in this section is to characterize the subset of {b⁼a⁼}, which equals to {(ab)⁼}, where a, b ∈ 𝒜.

Theorem 6. Let 𝒜 be any associative ring and let a, b ∈ 𝒜. Then,

where a₁ = ag_aa ≤⁻ a and b₁ = bg_bb ≤⁻ b.

Proof. Note that outer inverse of any element is a regular element in the ring, and if x is any regular element in 𝒜, then x ∈ x𝒜 ∩ 𝒜x.

Now consider any g_ab ∈ {(ab)⁼}. As noted earlier in the Lemma 5, we have g_ab = g_ababg_ab in the form of g_bg_a, where g_b = g_aba ∈ {b⁼} and g_a = bg_ab ∈ {a⁼}. Now,

The proof of g_a𝒜 = b₁𝒜 is similar. From the definition of a₁, it is verified that g_a is a generalized inverse of a₁. Further, we get a₁g_a = ag_a and g_aa₁ = g_aa, proving that a₁ ≤⁻ a. b₁ ≤⁻ b is similarly proved.

Now to prove (9) and the theorem, consider g_a ∈ {a⁼} and g_b ∈ {b⁼} satisfying 𝒜g_b = 𝒜a₁ or g_a𝒜 = b₁𝒜. Now consider the case of 𝒜g_b = 𝒜a₁, whereas the other case similar. Since g_b and a₁ are regular elements, we have g_b, a₁ ∈ 𝒜g_b = 𝒜a₁ and therefore, a₁bg_b = a₁ and g_aabg_b = g_aa. Now, using the same we get

proving that g_bg_a ∈ {(ab)⁼}.

Remark 1. Note that the set {g_bg_a : g_a ∈ {a⁼}, g_b ∈ {b⁼}, such that 𝒜g_b = 𝒜a₁ or g_a𝒜 = b₁𝒜} does not determine exact class of pairs of outer inverses (g_a, g_b) of a and b whose product determine {(ab)⁼}. It may be noted in each of the cases (i) g_a𝒜 ⊂ b₁𝒜 (ii) b₁𝒜 ⊂ g_a𝒜 (iii) 𝒜a₁ ⊂ 𝒜g_b (iv) 𝒜g_b ⊂ 𝒜a₁, we get

Therefore, determining the exact class of (g_a, g_b) determining {(ab)⁼} remains to be a problem to probe.

Let 𝒜 be an associative ring with involution ‘*’. Then, we have the following corollary.

Corollary 7. Given elements a, b ∈ 𝒜, if g_a and g_b are reflexive g-inverses of a and b, respectively, satisfying any of the cases (a) g_a𝒜 ⊂ b𝒜 (b) b𝒜 ⊂ g_a𝒜 (c) 𝒜a ⊂ 𝒜g_b (d) 𝒜g_b ⊂ 𝒜a, then $g_b{g}_a\in \left\{{(ab)}_r^{-}\right\}$. Further, we have the following.

Proof. In the cases of (a), (b), (c), and (d) we have bg_bg_a = g_a, g_aab = b, abg_b = a, and g_bg_aa = g_b, respectively. Now, by direct substitution it is easily verified that g_bg_a is a reflexive generalized inverse of ab.

If g_a ∈ a^{1,2,3} and the condition (a) or (c) is satisfied, then abg_bg_a = ag_a proving that g_bg_a ∈ {(ab)^{1,2,3}}. Hence, (i) is proved.

Similarly, if g_b ∈ a^{1,2,4} and the condition (b) or (d) is satisfied, then we have g_bg_aab = g_bb proving g_bg_a ∈ {(ab)^{1,2,4}} and (ii).

Proof of (iii) and (iv) are consequences of (i) and (ii), respectively, since $a^{†}\underset{sp}{\simeq }{a}^{*}$.

Proof of (v) is an immediate consequence of (iii) and (iv).

Let A and B be any two complex matrices with AB defined. According to Corollary 7 (v), we have (AB)^† = B^†A^† if ℛ(A) = ℛ(B*), where ℛ(X ) denotes the row space of X. However, the converse need not be true. For example, if

then ${(AB)}^{†}=\left[\begin{array}{l}\begin{array}{l}0\\ 1\end{array}\hfill \end{array}\begin{array}{l}\begin{array}{l}0\\ 0\end{array}\hfill \end{array}\right]$ = B^†A^†. But ℛ(B*) ≠ ℛ(A).

Left-star and right-star orders

In the present section, we extend the notions of left-star and right-star orders to the case of elements from an associative ring with involution. These notions were originally defined in the matrix case by Baksalary and Mitra in 12. As noted earlier, Drazin proved that the star order is a partial order on a proper *-semigroup and hence on a proper *-ring (see 11).

Following the matrix case, we shall now define the left-star and right-star orders in the context of associative rings with involution.

Definition 8. For a, b ∈ 𝒜, we say that a is dominated by b under left-star order, and denote a ${\le }_l^{*}$ b if a*a = a*b, a ∈ a𝒜 and a𝒜 ⊆ b𝒜. We say that a is dominated by b under right-star order, and denote a ${\le }_r^{*}$ b if aa* = ba*, a ∈ 𝒜a and 𝒜a ⊆ 𝒜b.

Baksalary and Mitra in 12 showed that the right-star and left-star orders are partial orders on the class of matrices over complex field. In the case of rings, clearly these relations are reflexive. Further, if a ${\le }_l^{*}$ b and b ${\le }_l^{*}$ c, then we have a*a = a*b = a*c, since a* = xb* for some x and b*b = b*c. Thus, the the relation ${\le }_l^{*}$ is transitive and preorder on ring 𝒜. Additionally, if a or b is left *-cancellable and a ${\le }_l^{*}$ b and b ${\le }_l^{*}$ a, then a = b. For the proof, consider the case of a being left *-cacellable. Then, a*a = a*b ⇒ a = b, since b = ax for some x and a ∈ a𝒜. Hence, we have the following theorem.

Theorem 9. Let 𝒜 be an associative ring with involution ‘*’. Then the relation ${\le }_l^{*}$ (${\le }_r^{*}$) is a partial order on the class of left *-cancellable (right *-cancellable) elements. In particular, if 𝒜 is a proper *-ring, then ${\le }_l^{*}$ and${\le }_r^{*}$ are partial orders on 𝒜.

Now, we shall proceed to prove some of the basic properties of left-star ordering. Analogous results for the right-star order can be obtained in the similar manner.

Theorem 10. Let 𝒜 be an associative ring and let a ∈ 𝒜 be left *-cancellable. Then the following statements hold.

Proof. Note that (i) follows from the properties of involution ‘*’.

From the hypothesis of (ii), we have a*a = a*p = pa and a ∈ a𝒜 ⊆ p𝒜. Therefore, a = pa = a*a proving that a is a projector. This proves (ii).

The equivalences of (a)-(f) in (iii) are easily verified using the properties of projectors.

Proof of (iv) follows from the statements (ii) and (iii).

Now, for a, b, c ∈ 𝒜^†, we make the following observations.

The reverse order law for Moore-Penrose inverse

The projectors play a prominent role in obtaining several equivalent conditions for the reverse order law for Moore-Penrose inverse. They have distinct properties associated with the star order, a few of which are discussed in 29. In this section, we shall note a few more interesting behaviors of projectors associated with star, left-star and right-star orders. An important observation in proving our main theorem is the following fact: For a projector p, we have pa ${\le }_r^{*}$ a (ap ${\le }_l^{*}$ a) if and only if pa ≤* a (ap ≤* a).

Lemma 11. Let 𝒜 be an associative ring with involution ‘*’ and let p be a projector. For any a ∈ 𝒜, the following are equivalent.

In particular, if a ∈ 𝒜a, then

Proof. Note that (i)⇒(ii), (ii)⇒(iii), and (iii)⇒(iv) are obvious.

Evidently, (pa)*(pa) = (pa)*a. Now, pa(pa)* = p(pa)a* = paa* = a(pa)*. This proves (iv)⇒(v).

Now, (14) follows from the definition of star order, equivalence of (i) and (v), and by noting that a ∈ 𝒜a implies that pa ∈ 𝒜pa, as p is idempotent.

Symmetrically, we have the following result.

Lemma 12. Let 𝒜 be an associative ring with involution ‘*’ and let p be a projector. For any a ∈ 𝒜, the following are equivalent.

In particular, if a ∈ a𝒜, then

Lemma 13. Let 𝒜 be an associative ring with involution ‘*’ and let a ∈ 𝒜. If p is a projector commuting with a, then pa ≤* a and ap ≤* a.

Proof. Follows by noting that the hypothesis implies statement (iii) of Lemma 11 as well as Lemma 12.

Now we prove the main theorem of the present paper. The equivalence of statements (i) and (vii) is discussed in 2 and in 21 and the equivalence of (i) and (vi) is proved in 30. However, our proof is in the general setting of Lemma 11 and Lemma 12, which use the properties of projectors.

Theorem 14. Let 𝒜 be a proper *-ring. For a, b ∈ 𝒜^†, the following are equivalent.

Proof. (i)⇒(ii) can be verified by direct substitution. Conversely, premultiplying a^†ab(b^†a^†a)a^†ab = a^†ab by a and postmultiplying (b^†a^†a)a^†ab(b^†a^†a) = b^†a^†a by a^†, we get abb^†a^†ab = ab and b^†a^†abb^†a^† = b^†a^†, respectively. Also, we have abb^†a^† = abb^†(bb^†a^†) and b^†a^†ab = (b^†a^†a)a^†ab. Hence, abb^†a^† and b^†a^†ab are Hermitian. Thus, we have (ab)^† = b^†a^†, proving (ii)⇒(i). This proves (i)⇔(ii).

The equivalence of the statements (iii)-(viii) will follow from the Lemma 11 and Lemma 12 noting that a^†a and bb^† are projectors. It remains to prove that (i)⇔(viii).

(i)⇔(viii): Evidently, (a^†ab)*a^†ab = b*a*(a^†)*a^† ab = (b*a^†a)b = (a^†ab)*b.

Premultiplying b*a* = b^†a^†ab(ab)* by abb*b, we get

Taking u = bb*a* and v = a^†abb*a*, we note that u*u = u*v = v*u = v*v = abb*a^†abb*a* and hence u = v, since ‘*’ is a proper involution. That is, bb*a* = a^†abb*a*. Therefore, we get

which proves that a^†ab ≤* b. Proof of other inequality is symmetric.

Conversely, we have that b^† and a^† are {1,3,4}-inverses of a^†ab and abb^†, respectively. Hence, b^†a^†ab and abb^†a^† are Hermitian. Also, premultiplying a^†abb^†a^†ab = a^†ab by a, we get abb^†a^†ab = ab.

Further, 𝒜 (a^†ab)* ⊆ 𝒜b* = 𝒜b^†, gives

Premultiplying by b^†(b^†)* and post-multiplying by a^†, we get

proving (i).

From (i)⇔ (ii) of Theorem 14, we have (a^†ab)^† = b^†a^†a and (abb^†)^† = bb^†a^† if and only if the reverse order law for Moore-Penrose inverse holds. Further from (i)⇔(viii) and (7), we have (ab)^† = b^†a^† if and only if (a^†ab)^† ≤* b^† and (abb^†)^† ≤* a^†. Hence, we have the following corollary.

Corollary 15. Let 𝒜 be a proper *-ring and let a, b ∈ 𝒜^†. Then the following are equivalent.

From Theorem 14, we have that a^†ab ${\le }_r^{*}$ b and abb^† ${\le }_l^{*}$ a if and only if the reverse order law (ab)^† = b^†a^† holds. The equivalence fails to exist if we interchange the left-star and right star orders. However, one of the implications is true and the same is proved in the following theorem.

Theorem 16. Let 𝒜 be a proper *-ring. Let a, b ∈ 𝒜 be such that a^† and b^† exist. Then,

Proof. We have, (a^†ab)*(a^†ab) = b*a*(a^†)*a^†ab = b*a^†ab = (a^†ab)*b. Further, taking u = a^†ab and v = bb^†a^†ab, we get u*u = u*v = v*u = v*v = b*a^†ab.

Thus,

which proves that a^†ab ${\le }_l^{*}$ b. Proof of other inequality is symmetric.

Remark 2. The converse of the above theorem need not be true in general. For example, if

then

so that

Clearly, A^†AB ${\le }_l^{*}$ B and ABB^† ${\le }_r^{*}$ A. But

From Lemma 11 and Lemma 12, we have that a^†ab ${\le }_r^{*}$ b implies a^†ab ≤* b and abb^† ${\le }_l^{*}$ a implies abb^† ≤* a. It is not true in general that a^†ab ≤* b whenever a^†ab ${\le }_l^{*}$ b and that abb^† ≤* a whenever abb^† ${\le }_r^{*}$ a. The following lemma gives a sufficient condition for this result to hold.

Theorem 17. Let 𝒜 be a proper *-ring and let a, b ∈ 𝒜^†. Suppose (ab)^† = b^†a^†. Then, a^†ab ${\le }_l^{*}$ b implies a^†ab ≤* b and abb^† ${\le }_r^{*}$ a implies abb^† ≤* a.

Proof. Since a^†ab𝒜 ⊆ b𝒜 , we have

as desired. The second implication can be proved similarly.

The proof of the following theorem is immediate from Theorem 14 and Lemma 13, noting that a^†a and bb^† are projectors commuting with b and a, respectively. This theorem shows that only two commutative relations out of four, given by Erdelyi in (Theorem 1¹) are sufficient for the reverse order law.

Theorem 18. Let 𝒜 be a proper *-ring and let a, b ∈ 𝒜 be such that a^† and b^† exist. If both a(bb^†) = (bb^†)a and b(a^†a) = (a^†a)b are true, then (ab)^† = b^†a^†.

Conclusions

In the present paper, the reverse order law for outer inverses is discussed. The notions of left-star and right-star orderings are extended to the case of associative rings with involution and several properties of the same are discussed. Also, necessary and sufficient conditions—in terms of star, left-star and right-star orders—for the reverse order law for Moore-Penrose inverse are given. The discussion on left-star and right-star partial orderings appears to help in further study of relation between star partial order and column space decomposition of matrices. Further, it has been noted in Theorem 10 that, in the class of all projectors the star, left-star, right-star and minus orders are all equivalent. The classification of elements of associative rings for which any two or more orders are equivalent seems to be an interesting problem to explore and the same will be presented in the subsequent papers.

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