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Regular Matrices in the Semigroup of Hall Matrices

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Regular Matrices in the Semigroup of Hall Matrices View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Regular Matrices in the Semigroup of Hall Matrices Han H. Cho* Department of Mathematics College of Education Seoul National University Seoul. Korea Submitted by Richard A. Brualdi ABSTRACT We study the regular matrices in the semigroup H, of Hall matrices (Boolean matrices with positive permanent). We study some necessary and sufficient conditions for a Hall matrix to be regular in H, in terms of idempotent matrices, adjoint matrices, and identifying permutation matrices. 1. INTRODUCTION Let p = (0, 1) be the B oo 1can algebra of order two with operations (+;):1+0=0+1=1+1=1~1=1&0+0=0*1=1~0=0~0 = 0 and an order < : 0 < 1. Then under these Boolean operations, the set B, of all n X n matrices over p (Boolean matrices) and the set H,, of all n X n matrices over p with positive permanent (Hall matrices) form multi- plicative semigroups. There have been many researches on such semigroup properties as primeness, regularity, and indices in B, and H,. In this paper we study the regular matrices in the semigroup H,(s) (= {A E B,, ( per A > s}) for each positive integer s. DEFINITION 1.1. Let S be a multiplicative semigroup, and let A be an element of S. A is regular in S if AGA = A for some G E S (G is called a * Supported by a grant from the Korea Research Foundation LINEAR ALGEBRA AND ITS APPLICATIONS 191: 151-163 (1993) 151 0 Eisevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010 0024-3795/93/$6.00 152 HAN H. CHO generalized inverse of A). A is semiinvertible in S if AGA = A and GAG = G for some G E S (G is called a semiinverse of A). For a Boolean matrix A E B,,, the Boolean rank of A is the smallest integer r for which there exist n X r and r X n Boolean matrices B and C providing the factorization A = B . C. The Boolean rank of a zero matrix is 0, and if the Boolean rank of A E B, is n, then A is called a rank-n matrix. Throughout this paper, we will regard the regular elements of N,,(s) as a generalization of the invertible elements in the semigroup R, of n X n real matrices. With this point of view in mind, we derive some properties of the regular matrices of H,,(s) by examining the well-known properties of the invertible matrices in R,,. Al so in this paper we show that many properties that hold for the rank-n regular matrices of B,, also hold for the regular matrices of H,(s). DEFINITION 1.2. Let A E B, be an n X n Boolean matrix. Then for each pair of integers i and j, Ai * and A *j denote respectively the i th row and the jth column of A, and Aij denotes the (i, j) entry of A. As usual, A’ denotes the transpose of A. For A and R in B,, R < A if and only if Rij < Ai. for all i and j. If R < A, then R is called a spanning submatrix of A (we a i so say that R is contained in A), and A - R denotes a Boolean matrix such that (A - R)i. = 1 if and only if A, = 1 and Rij = 0. The permanent per A of A is t 1:e number of elements in S,, where S, = {P E S, ( P < A) and S, is the set of all n X n permutation matrices (S, also denotes the set of all permutations on { 1, . , n}). Finally, both 1Al and a( A) denote the number of ones of A E B,L, and A is called a J-matrix and denoted by J,, if a( A) = n’. Consider the following commutative diagram: Here, N, denotes the monoid of n X n nonnegative matrices, R, denotes the monoid of n x n doubly stochastic matrices, and T,, denotes the monoid of n X n Boolean matrices with total support (Boolean matrices that can be expressed as a sum of permutation matrices). Finally L denotes the canonical inclusion map, and r denotes the support map that sends a nonnegative REGULAR HALL MATRICES 153 matrix to a Boolean matrix in such a way that for each (Y E N,,, the (i, j> entry m( a>,] of rr(a) is 1 if and only if crij > 0 for each i and j. Since 1 . 1 = 1 in P and the product of any two positive numbers is positive in the real number system, Z-((Y) = rTT(/3)r(Y) when (Y = PY. Therefore r is a semigroup homomorphism, and using this semigroup homomorphism r we can compare the regular elements of each semigroup in the diagram as follows. THEOREM 1.3. Let a be an element of N,, (respectively Cl,) and let A be T(a). Then (1) Zf a is rqqdar in N,, (respectively a,), then A is regular in B,, (respectively T,, ). (2) A is regular in B,, if and only if ( AtA“A’ )” (A” is Jn - A) is the largest generalized inverse of A in B,, (3) (Y E s1,, is regular in R,, if and only if cxcx$2 = cr. Proof. Refer to [2] for (1). Refer to Schein [9] for (2). Finally, refer to Robinson [8] for (3). W Consider the matrices 1 1 and C = [ 1 1 1 Then Y is not regular in Nz, but C is regular in B,. So the converse of (1) in Theorem 1.3 does not hold. By (2) and Corollary 4.4 we can see that for A EH,,, A is regular in H,, iff ( AtAZAf)“ = P’APt for some P E S, iff ( A’AcA’)c E H,. Consider the matrices 1 0 0 1 1 0 0 0 ^=I0 0 1 11’ L 0 0 10’ * 1100 1 0 111 1 y=o i 0 010 0 0 I Then A E H,, and A is regular in B, since AGA = A and GAG = G hold. But A is not regular in H,, by Theorem 4.3, since the first row A, * of A contains only one row of A even though 1A, + 1 = 2. Therefore there is a Hall matrix A E H, of rank less than n such that A is regular in B, but not in H,. Note that for a rank-n Boolean matrix or a prime Boolean matrix A E B,, A is regular in B,, if and only if A is regular in H, (cf. [2]). 154 HAN H. CHO 2. REGULAR MATRICES AND IDEMPOTENT MATRICES In this section we characterize the regularity of a Hall matrix A E H, in terms of idempotent matrices. Then for each positive integer s, we compare the regular matrices in H,(s) and T, with the regular matrices in H,. DEFINITION 2.1. For A E H,, A is called an idempotent matrix if AA = A. For any Boolean matrices A and l? in B,, we say A is (permuta- tionally) equivalent to B if B = PAQ f or some permutation matrices P and Q. Also, if B = PAPt for some permutation matrix P, then we say A is (permutationally) similar to B. LEMMA 2.2. Let A and G be Hall matrices such that AGA = A. Then for each P E S,, APA = A, and Pt E S,. Proof. For each P E S,, AGA = A means APA Q A and 1APAl < 1Al. Since AP contains a permutation matrix, I( AP) Al > 1A( holds. Thus we have APA = A and PAPA = PA for any P E S, . Since PAPA = PA and (P - S,) . (P * S,> E P * S,, P - S, is a subsemigroup of the symmetric group S, (P . S, is {P * Q 1 Q E S,)). So P * S, is a group. Thus PQ = T for some Q E S,, and P ’ is in S,. THEOREM 2.3. Let A be an n X n Hall matrix. Then the following statements are all equivalent: (1) A is regular in H,. (2) A has a unique semiinverse in H,. (3) A is per-mutationally equivalent to an idempotent matrix. (4) A is regular in H,(s) if A E H,(s). Proof. (1) --) (2): Let S = {X E H, ( AXA = A}, and let G be the Boolean sum AXES X. Then AGA = A(C,,,X)A = C,,,(AXA) = A. Therefore G is the largest generalized inverse of A in H,. Since A(GAG) A = (AGA)GA = AGA = A, we obtain GAG ( G. Note that AGA = A and GAG < G imply IGI ( IAl and IAl < IGI respectively. Thus (Gl = (A( and GAG = G. Hence G is also the largest semiinverse of A in H,. Now consider any semiinverse H of A in H,. Since A(G + H)A = A and G is the largest generalized inverse of A, we obtain H < G. Since I Al = (Cl and HAH = H, we have JHI <IAl and IA1 6 IHI. Thus IAl = JHI = ICI and H = G. Therefore there exists a unique semiinverse of A in H,. REGULAR HALL MATRICES 155 (2) + (3): By L emma 2.2, AGA = A implies APA = A for any P E So. So ( APA)P = AP and AP is an idempotent matrix of H,. Thus A is permutationally equivalent to an idempotent matrix.
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