Literally Idempotent Languages and their Varieties - Two Letter Case

A language L ⊆ A* is literally idempotent in case that ua2v ∈ L if and only if uav ∈ L, for each u, v ∈ A*, a ∈ A. We already studied classes of such languages closely related to the (positive) varieties of the famous Straubing-Therien hierarchy. In the present paper we start a systematic study of literal varieties of literally idempotent languages, namely we deal with the case of two letter alphabet. First, we consider natural canonical expressions for such languages. Secondly, we describe all possible classes of the form where is a literal variety of literally idempotent languages. At the end we consider also positive literal varieties of literally idempotent languages.