JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 3, Pages 559 - 575
(December 2004)
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ON
MEET MATRICES WITH RESPECT TO REDUCED, EXTENDED
AND EXCHANGED SETS
Ismo Korkee (Finland) and Pentti Haukkanen (Finland)
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Abstract: Let
be
a meet-semilattice and let
be a subset of
P. The meet matrix
on S
with respect to a function
is defined as
There are in
the literature applicable formulae for
and
when S
is a meet-closed set (i.e.,
for all i, j).
In
this paper we extend these formulae by
providing calculation formulae for
and
in terms of
meet-closed subsets and supersets of S.
We compare the effectivity of these methods
based on reduction and extension of the set S
to the effectivity of row-reduction.
As
applications we first combine our reduction
and extension methods for calculating
and
by exchanging
the "difficult" elements of S
for "easy" ones. Second, we obtain
known formulae for
and new
formulae for
where S
is an a-set (i.e.,
for all
Next we give
new formulae for the determinant and the
inverse of the meet matrix
on two sets X
and Y, where
The methods of
this paper are also appropriate for join
matrices on join-semilattices. As special
cases these results hold also for GCD and LCM
matrices. |
Keywords and phrases: meet-semilattice, meet
matrix, determinant, inverse matrix, partitioned
matrix, join matrix, GCD matrix, LCM matrix. |
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