A Sufficient Condition for Hanna Neumann Property of Submonoids of a Free Monoid

Using automata-theoretic approach, Giambruno and Restivo have investigated on the intersection of two finitely generated submonoids of the free monoid over a finite alphabet. In particular, they have obtained Hanna Neumann property for a special class of submonoids generated by finite prefix sets. This work continues their work and provides a sufficient condition for Hanna Neumann property for the entire class of submonoids generated by finite prefix sets. In this connection, a general rank formula for the submonoids which are accepted by semi-flower automata is also obtained. Introduction Howson proved that the intersection of two finitely generated subgroups of a free group is finitely generated [7]. In 1956, Hanna Neumann improved the result that if H and K are finite rank subgroups of a free group, then r̃k(H ∩K) ≤ 2r̃k(H)r̃k(K), where r̃k(N) = max(0, rk(N)− 1) for a subgroup N of rank rk(N). Further, Neumann conjectured that r̃k(H ∩K) ≤ r̃k(H)r̃k(K), (⋆) which is known as Hanna Neumann conjecture [12]. In 1990, Walter Neumann proposed a stronger form of the conjecture called strengthened Hanna Neumann conjecture (SHNC) [13]. Meakin and Weil proved SHNC for the class of positively generated subgroups of a free group [9]. The conjecture has recently been settled by Mineyev (cf. [10, 11]) and announced independently by Friedman (cf. [3, 4]). In contrast, it is not always true that the intersection of two finitely generated submonoids of a free monoid is finitely generated. It appears that the intersection problem for submonoids of free monoids is much more complex than the analogous problem for subgroups of free groups. In particular, Hanna Neumann property for submonoids of a free monoid is of special interest. Two finitely generated submonoids H and K of a free monoid are said to satisfy Hanna Neumann property (in short, HNP), if H and K satisfy the inequality (⋆). There are several contributions in the literature to study the intersection of two submonoids of a free monoid. In 1972, Tilson proved that the intersection of free submonoids of the free monoid over a finite alphabet is free [16]. In connection to HNP, Karhumäki obtained a result for submonoids of rank two of the free monoid over a finite alphabet. In fact, 1991 Mathematics Subject Classification. 68Q70, 68Q45, 20M35.