Malaya Journal of Matematik, Vol. 8, No. 4, 1691-1693, 2020 https://doi.org/10.26637/MJM0804/0062 Williams meet matrices base b on A-sets N. Elumalai1 and R. Kalpana2* Abstract In this study Williams meet matrices base b = 3 on A-sets are considered to study the results based on structure theorem. The determinant and inverse of the matrix are also found. In addition the matrix is expressed in terms of the other two matrices. Keywords Meet matrices, Williams meet matrices base b, A-sets. AMS Subject Classification 15A09, 03G10. 1,2P.G. and Research Department of Mathematics, A.V.C. College (Autonomous)(Affiliated to Bharathidhasan University,Trichy), Mannampandal, Mayiladuthurai-609305, Tamil Nadu, India. *Corresponding author: [email protected] Article History: Received 24 July 2020; Accepted 17 September 2020 c 2020 MJM. Contents 2. Preliminaries In this section, we recall some definitions which are very 1 Introduction......................................1691 useful to this work. 2 Preliminaries.....................................1691 1. If S1 and S2 are nonempty subsets of P then S1 u S2 = 3 Theorems on Williams number base b = 2. 1691 [x ^ y=x 2 Sl;y 2 S2;x 6= yg where u is a binary opera- 4 Results on Williams meet matrices base b on A-sets tion. 1692 2. S = fx1;x2;::::::::::::xng ⊂ P with xi < x j ) i < j 5 Conclusion.......................................1692 and let A = fal;a2;::::::::::::an−1g be a multichain References.......................................1692 with a1 ≤ a2 ≤ :::::::::: ≤ an−1 The set S is said to be an A -set if fxkg u fxk+1;::::::;xng = fakg for all k = 1;2;::::::;n − 1. 1. Introduction 3. A matrix (S) of size n×n; where (S) f i j = f (xi ^ x j); Williams number base b are numbers of the form (b − is called the meet matrix on S with respect to f . 1)bn − 1; for integers b ≥ 2 and n ≥ 1: The meet matrices are defined on a set of positive integers S = fx ;x ;x ;:::::::::;x g: 4. Williams number base b is a natural number of the form 1 2 3 n n To define these matrices a complex valued function f from (b − 1)b − 1 for integers b ≥ 2 and n ≥ 1. P ! C is considered where P is a locally finite lattice (P;_;^). 5. Williams meet matrix base b is defined as a matrix (S) f Also S is considered as the subset of P. The meet matrix is xi^x j of size n × n, where (S) f i j = (b − 1)b − 1: denoted as (S) f The elements of these matrices are denoted as ((S) f )i j = f (xi ^x j). Haukkanen [2] has shown the method to find the determinant and inverse of these meet matrices. The 3. Theorems on Williams number base generalizations of GCD and LCM matrices are obtained in [1]. b = 2. The conjecture of Beslin and Ligh on greatest common divisor Theorem 3.1. If b and n are natural numbers such that bn −1 matrices as Meet matrices is studied in [3]. In this study we is prime, then b = 2 or n = 1. are going to study the nature of Williams Meet matrices base n n b with the special case of b = 2: Also find the determinant Proof. b ≡ 1(modb − 1) ) b ≡ 1(modb − 1): ) b − 1 ≡ and inverse of Williams meet matrices base b on A-sets with 0( mod n−1) Hence b−1 j bn −1: But bn −1 is prime, hence the special case of b = 3. b − 1 = bn − 1 or b − 1 = ±1: From the first value b = bn ) Williams meet matrices base b on A-sets — 1692/1693 x b = 0;1 (which is a contradiction, as neither 1 nor 0 is prime) f (x) =2 · 3 − 1 = f1(x); or n = 1. From the second value, b = 2 or b = 0. If b = 0 we 2 n ( f1 (a1)) x get 0 − 1 = 0 − 1 = −1 which is not prime. ) b = 2. f2(x) = f1(x) − = 2 · 3 − 6; ( f1 (x1) 2 ( f2 (a2)) x f3(x) = f2(x) − = 2 · 3 − 6 ( f2 (x2) Theorem 3.2. If (b − 1)bn − 1 when b = 2 is prime, then n is f1 (x1) =5; f2 (x2) = 12; f3 (x3) = 48: prime. ) det(S) f = f1 (x1) f2 (x2) f3 (x3) = 2880: Proof. If n is composite, then n = pq with p and q > 1, Hence determinant of Williams meet matrix base b = 3 on b =2 ) 2n − 1 = 2pq − 1 = (2p)q − 1 A-sets is calculated. T Now (S) f is expressed as (S) f = M DM =(2p − 1) (2p)q−1 + (2p)q−2 + ::: + 2p + 1 : 2 1 1 1 3 n n ) 2 − 1 is composite. By contrapositive, if 2 − 1 is prime M = 4 0 1 0 5 and D = diag(5;12;48) then n is prime. Hence the proof. 0 0 1 fi(ai) by using (M)i j = for all i < j and fi(xi) n Theorem 3.3. Every prime p that divides (b−1)b −1 when D = diag( f1 (x1); f2 (x2);::::::; fn (xn)) b = 2 must be 1 plus a multiple of 2n if n is an odd prime. The result is true even when 2n − 1 is prime. from structure theorem [5]. By usual calculations it can be T −1 proved that M DM = (S) f . The value of S f is also cal- p−1 Proof. Using Fermat’s little theorem, p is a factor of 2 −1: −1 T culated from S f = NDN using result from [5]. Now p is a factor of 2n − 1; for all positive integers a; p is na also a factor of 2 − 1: As n is prime and p is not a factor 2 1 −1 −1 3 1 of 2 − 1;n is also the smallest positive integer y such that p N = 0 1 0 ; y 4 5 is a factor of 2 − 1: For all positive integers y, p is a factor 0 0 1 of 2y − 1 if and only if n is a factor of y. As p is a factor of 2p−1 − 1;n is a factor of p − 1so p ≡ 1(modn): Now p is a and n factor of 2 − 1; which is odd ) p is odd ) p ≡ 1( mod 2n): 2 73 1 1 3 240 − 12 − 48 Hence the proof. 1 1 1 −1 1 1 D = diag ; ; ) S f = 4 − 12 12 0 5: 5 12 48 1 1 − 48 0 48 Thus the results of meet matrices on A-sets [5] are verified for 4. Results on Williams meet matrices Williams meet matrices base b = 3 on A-sets. base b on A-sets 5. Conclusion 1. Williams meet matrices base b on A-sets satisfies struc- ture theorem [5]. In this paper we have proved some theorems on Williams number base b = 2: The Williams meet matrices base b = 3 on 2. The determinant of this matrix is obtained by using the A-sets are considered to prove results from structure theorem. recursion formula. Other Williams meet matrices with different value of base b on A-sets may be considered for future study. Also some other 3. The inverse of this matrix is also found by using these type matrices and join matrices can also be included in the results [5]. study. We prove these results by considering the following example. References We consider b = 3. Then the Williams number base becomes [1] (b − 1)bn − 1 = 2 · 3n − 1: S. Beslin and S. Ligh, GCD-closed sets and the deter- x ^x The Williams meet matrix base b is ((S) f )i j = 2:3 i j − 1: minants of GCD matrices, Fibonacci Quart., 30(1992), Consider S = f1;2;3g then A = f1;1g. 157–160. [2] P.Haukkanen, On meet matrices on posets, Linear Alge- 2 5 5 5 3 bra Appl., 249(1996), 111–123. [3] (S) = 4 5 17 5 5 P. Haukkanen, J.Wang and J. Sillanpa¨a,¨ On Smith’s deter- 5 5 53 minant, Linear Algebra Appl., 258(1997), 251–269. 1692 Williams meet matrices base b on A-sets — 1693/1693 [4] I. Korkee and P. Haukkanen, On meet and join matri- ces associated with incidence functions, Linear Algebra Appl., 372(2003), 127–153. [5] I. Korkee, On meet and join matrices on A-sets and related sets, Notes on Number Theory and Discrete Mathematics, 10(3)(2004), 57–67. ????????? ISSN(P):2319 − 3786 Malaya Journal of Matematik ISSN(O):2321 − 5666 ????????? 1693.