The Join Levels of the Trotter-Weil Hierarchy Are Decidable

The variety DA of finite monoids has a huge number of different characterizations, ranging from two-variable first-order logic FO to unambiguous polynomials. In order to study the structure of the subvarieties of DA, Trotter and Weil considered the intersection of varieties of finite monoids with bands, i.e., with idempotent monoids. The varieties of idempotent monoids are very well understood and fully classified. Trotter and Weil showed that for every band variety V there exists a unique maximal variety W inside DA such that the intersection with bands yields the given band variety V. These maximal varieties W define the Trotter-Weil hierarchy. This hierarchy is infinite and it exhausts DA; induced by band varieties, it naturally has a zigzag shape. Kufleitner and Weil have shown that it is possible to ascend the corners of the Trotter-Weil hierarchy in terms of Mal’cev products with definite and reverse definite semigroups and also in terms of so-called rankers. Using a result of Straubing, one can climb up the intersection levels of the Trotter-Weil hierarchy in terms of weakly iterated block products with J -trivial monoids. In their paper, Trotter and Weil have shown that the corners and the intersection levels of this hierarchy are decidable. In this paper, we give a single identity of omega-terms for every join level of the Trotter-Weil hierarchy; this yields decidability. Moreover, we show that the join levels and the subsequent intersection levels do not coincide. Almeida and Azevedo have shown that the join of R-trivial and L-trivial finite monoids is decidable; this is the first non-trivial join level of the Trotter-Weil hierarchy. We extend this result to the other join levels of the Trotter-Weil hierarchy. At the end of the paper, we give two applications. First, we show that the hierarchy of deterministic and codeterministic products is decidable. And second, we show that the direction alternation depth of unambiguous interval logic is decidable.