Due to the involution on any inverse semigroup, this will also be isomorphic to the lattice of right

congruences.

Then the entries [a.sub.i](x) and [b.sub.i](x) of the matrix A(x) and of the matrix B(x), obtained as a result of transformation, satisfy the

congruences (16)-(18).

In view of

congruences (3.37) and (3.39), we arrive at (3.9).

In [1], Saikia and Boruah considered the

congruence identities for [c.sub.sl](n) by employing the following lemma.

Particularly, b adic expension of an integer N and its

congruence modulo b was used to characterise a given integer regarding its divisibility rule.

Recall that a monoid

congruence is an equivalence relation [equivalent to] which is left and right compatible with the product; in other words, for any monoid elements a, b, c, d, if a [equivalent to] b and c [equivalent to] d then ac [equivalent to] bd.

Some single-country longitudinal studies have tried to explain opinion-policy

congruence using variables that do vary in the medium run, like the party in power or the size of the majority in parliament.

Miezaki,

Congruences on the Fourier coefficients of the Mathieu mock theta function.

Assume now that f + g < n - 1 and let us find the solutions of the

congruence [([p.sup.2] + rq).sup.2] [equivalent to] 1 (mod [2.sup.n]).

For the definitions and generalities about systems of conservation laws and their correspondence with

congruences of lines, we refer to [AF01], see also [DM05].

If ~ is a

congruence on a KU-algebra G, then [C.sub.0] = {x [member of] G | x ~ 0} is an ideal of G.

Ojah, "Analogues of Ramanujan's partition identities and

congruences arising from his theta functions and modular equations," Ramanujan Journal, vol.