@InProceedings{	  KufleitnerLauser12mfcs,
  title		= "The join levels of the {T}rotter-{W}eil hierarchy are
		  decidable",
  author	= "Manfred Kufleitner and Alexander Lauser",
  booktitle	= "Mathematical Foundations of Computer Science,
		  International Symposium (MFCS) 2012, Conference
		  Proceedings",
  year		= "2012",
  pages		= "603--614",
  volume	= "7464",
  series	= "Lecture Notes in Computer Science",
  publisher	= "Springer",
  abstract	= "The variety DA of finite monoids has a huge number of
		  different characterizations, ranging from two-variable
		  first-order logic $\mathrm{FO}^{2}$ 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.
		  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.",
  doi		= "10.1007/978-3-642-32589-2_53"
}