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Single-Element Extensions of Matroids

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Single-Element Extensions of Matroids JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 69B, Nos. 1 and 2, January-June 1965 Single-Element Extensions of Matroids Henry H. Crapo Assistant Professor of Mathematics, Northeastern University, Boston, Mass. (No ve mbe r 16, 1964) Extensions of matroids to sets containing one additional ele ment are c ha rac te ri zed in te rms or modular cuts of the lattice or closed s ubsets. An equivalent characterizati o n is given in terms or linear subclasse s or the set or circuits or bonds or the matroid. A scheme for the construc ti o n or finit e geo· me tric lattices is deri ved a nd th e existe nce of at least 2" no ni somorphic matroids on an II- ele ment set is established. Contents Page 1. introducti on .. .. 55 2. Differentials .. 55 3. Unit in crease runcti ons ... .... .. .... ........................ 57 4. Exact diffe re ntia ls .... ... .. .. .. ........ " . 58 5. Sin gle·ele me nt exte ns ions ... ... 59 6. Linear subclasses.. .. .. ... .. 60 7. Geo metric la ttices ................... 62 8. Numeri ca l c lassification of matroid s .... 62 9. Bibl iography.. .. ............... 64 1. Introduction I of ele ments Pi of a lattice L, in whic h Pi covers Pi - t for i = 1, . .. , n, is a path (of length n) from x to y. In order to facilitate induc tive proofs of matroid 2 theore ms, we shall set forth a characte rization of single­ A step is a path of length J. element extensions of matroid structures. Our tech­ The partial order of the lattice induces an order niques are those developed in our recent paper, on the set of steps of the lattice. If [x, y] and [u , w] " Lattice Differentials and the Theory of Combinatorial are steps of a lattice L, we say [x, y] ~ [u , w] if and Indepe nde nce". Sufficient material is included in only if x = y" u and yv u = w. If, in a modula r the next three sec ti ons to support the proofs of the lattice, a ste p [u , w] cove rs a s tep [x, y] in this ordering, extension theore ms. The reader is referred to the then [u, x] and [y, w] are also steps. above pape r for de tails. Definition: A func ti on R d efin ed for all steps A matroid, defin ed on a finite set, may be thought [x, y] of a finite modular lattice L, a nd ta king values of as a structure de fin ed on a lattice, i. e., on the in the tw o-ele me nt lattice 3 {D, I}, is a differential if and only if it is Boolean algebra of all subsets of the set. Not all the properties of Boolean algebras are use d in the (a) Projective on ste ps: If s teps [x, y] and [u , w] are proof of matroid theorems, however. W e have in the order [x, y] ~ [u, w], the n R[x, y] ~ R[u, w] . chosen modular lattices as the proper domain for (b) Subadditive on steps: If a step [u, w] covers a our presentation of the material needed for extension step [x , y] the n R[x, y] ~ R[x , u] v R[u, w]. theory. The local character of differe ntials may be repre­ sented pictorially. A local graph is any assignment 2. Dilierentials of values D or 1 to the four steps of an interval [X" y, xvy] in a finite modular lattice L, where the elements y An element is said to cover an ele ment x in a x and y cover x A y and are thus covered by x v y. lattice L if x < y, and for any eleme nt z, x < z ~ y, PROPOSITION: An assignment of values D and 1 to implies z = y_ A lattice L is modular if, for all ele­ the steps of a finite modular lattice L determines a ments x and y in L, x covers X" Y if and only if x v y differential if and only if aLL local graphs are of the covers y. A sequence p : x = Po < Pt < .. < p" = y foLLowing five types: zero, mixed, prime, one, or inexact. (These types of local graph are defined as fi gure 1. We indicate steps on which R = 1 by double line s, ' The s tate m{' nt ,If th e e xtension thelJn' 1ll a nd th t., c lassifi ca ti' lll of malroids are taken frllm tilt' aut hor's Ph.D. thl." s is. entitled " O n lilt' Tlu!ory of C Ollibi nalllrial Ind"IH'ndc nce", R = D by single lines.) wri tt en und.· .. I lll' s upc r vi :; j.,n III' P rjd't'ssur C ia n-Carl' l Hilla. alld pn.:: scntl.:d lu Ihe Mass . In ~ 1. of Tt'chnoiog y. May 1964. \'\' 1' UI"I.: g rat eful 10 Jac k Edmunds. Nat iona l i3urcau III' SI <.I nclHrds. w h' l 1)llinl('(1 Ilu l Ill«' :..; ig nificancl' Ilf tlw !·w IIWtlft: lns a nd 1' llc11ur<1gt·d us in Il1I'ir p rt.'se tll <1 liull, :1 DifTt'I"l 'nli <1l !' uf highe r order lIlay Ut· lIiJlailwd lJ y <1 I1 (1wing tilt' rangt· of ti lt' difft'l"t'nl ia l 1. 11. W hilrlt:y. Anwr. .J .• 19 3,1) . 10 Le a lallie(' Illore ge ncrailhan {a , I }. 55 _J path exists. Assume the truth of the statement for n-l, and let L have height n. Since L is modular, either WVpn_I=UVp"_1 and W" pn- I covers U" pn- I, or else WV Pn_1 covers UVPn _1 and W"Pn- I= U"p"- I' In the first instance, we have the ordering of steps ooo¢o [u " Pit- I, w" Pn - d ,,;; [u, w], so R[u" Pn - I, W" Pn- I] = 1 ZERO Mi X ED PRIME ON E INE XACT in the lattice interval [0, Pn- d of height n - 1. This step cannot coexist with the portion of the path P from FIGURE 1 o to Pit _I. In the second instance, choose a path q from u to Pit - I, and let} be the greatest index for whic h PROOF: The projective and sub additive properties of R[/li , w v qjl=1. By subadditivity of R on steps, differentials eliminate all local graphs except the five R[qj, qj+d = 1, where [qj, qj+d is also a step in the enumerated above. Conversely, if R is a function interval sublattice [0, Pn - I] of height n-l. on steps with values in the lattice {O, 1}, and if all PROPOSITION: Closures Formed from Differentials: local graphs of R are zero, mixed, prime, one, or in· Given a differential R on a finite modular lattice L, exact, the subadditive property follows, and the pro· CI(x) = sup {y; 3: path from x to y along which R = O} jective property is established, for steps [x , y] and =inf {z; z closed, x ";; z} [u, w], with [x, y] ,,;; [u , w], as follows. Let x = Po < PI < .. < PI/=u be a path from x to u, and let q=p vy is a closure operator on the lattice L, the differential be the projected path y=povy < pl vy< ... < pnvy of which is R. = w from y to w. If R[x, y] < R[u, w], then R[x, y]= O PROOF: If an element x is connected to two elements and R[u, w]= 1. Let Pi be the greatest element such YI and Y2 by paths along which R = 0, every step in a that R[Pi, q;] = O. Then R[Pi+l, qi +I] = 1. Such a path from YI to YI V Y2 exceeds, in the ordering on local graph pi, qi, pi+l, qi +1 is neither zero, mixed, steps, some step in the path from x to Y2. Thus prime, nor inexact. R = 0 on the extended path from x to YI V Y2, so A differe ntial R is termed exact if and only if it has YI V Y2 ,,;; Cl(x). If a path exists from x to y along no inexact local graph. Exac t differe ntials are the which R = 0, then R must have value 0 on all steps of subject of section 4, below. the lattice interval [x, y], as was shown in the proof The three principal sources of differentials are of the previous proposition. Thus the closure Cl(x) closure operators, join-homomorphisms, and finite may be characterized as the maximum element y lattices. We describe these in turn. such that R = 0 everywhere on the lattice interval If Cl is a closure operator (x ";; Cl(x), and x ,,;; Cl(y) [x, y] . This element Cl(x) must be closed, for other­ implies Cl(x) ..s; Cl(y» on a finite modular lattice L, wise a path along which R = 0 would exist from x to an element covering Cl(x). R[x, y] = 1 if Cl(x) < Cl(y) The set of closed e lements of the lattice L is closed = 0 otherwise with respect to the lattice operation inf If x and y are closed elements, and an element u covers X" y, defines a differential R on the steps of the lattice L. then u 1 x or u 1 y.
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