  One matrix structure in the area of monotone Boolean functions is defined
here. Some of its combinatorial, algebraic and algorithmic properties are
derived. On the base of these properties, three algorithms are built. First of
them generates all monotone Boolean functions of $n$ variables in lexicographic
order. The second one determines the first (resp. the last) lexicographically
minimal true (resp. maximal false) vector of an unknown monotone function $f$
of $n$ variables. The algorithm uses at most $n$ membership queries and its
running time is $\Theta(n)$. It serves the third algorithm, which identifies an
unknown monotone Boolean function $f$ of $n$ variables by using membership
queries only. The experimental results show that for $1\leq n\leq 6$, the
algorithm determines $f$ by using at most $m.n$ queries, where $m$ is the
combined size of the sets of minimal true and maximal false vectors of $f$.
