Boolean matrices ... neither Boolean nor matrices

Abstract

Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

A semantic construction of two-ary integers

“…By 5.4 (B) the sum and ∅ form a monoid that we call the prime analytic monoid. (Recall that by 5.0 in this monoid the composition of u with a is a + u, see also [18].) On the contrary, we call sum monoid the reversed one where the composition of u with a is u + a.…”

“…Contrary to [12], see [18], we consider functional composition as the restriction of relational composition, namely f · g is "the composition of g and f " and…”

“…where we call function M the increment matrix, and the term matrix product : 18] shows the necessity of such reversed readings.) As 1.7 (B) will show, is the composition of a monoid with unit i U .…”