One of the prominent mathematical features of natural language is the prevalence of "upward" and "downward" inferences involving determiners and other functional expressions. These inferences are associated with negative and positive polarity positions in syntax, and they also feature in computer implementations of textual entailment. Formal treatments of these phenomena began in the 1980's and have been refined and expanded in the last 10 years. This paper takes a large step in the area by extending typed lambda calculus to the ordered setting. Not only does this provide a formal tool for reasoning about upward and downward inferences in natural language, it also applies to the analysis of monotonicity arguments in mathematics more generally.

We study finite state transduction of automatic and morphic sequences. Dekking [4] proved that morphic sequences are closed under transduction and in particular morphic images. We present a simple proof of this fact, and use the construction in the proof to show that non-erasing transductions preserve a condition called α-substitutivity. Roughly, a sequence is α-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue α. Our results culminate in the following fact: for multiplicatively independent real numbers α and β, if v is a α-substitutive sequence and w is an β-substitutive sequence, then v and w have no common non-erasing transducts except for the ultimately periodic sequences. We rely on Cobham's theorem for substitutions, a recent result of Durand [5]. * This is an extended version of our paper [9] presented at Developments in Language Theory 2014. This extended version contains examples and additional remarks. Related workDurand [5] proved that if w is an α-substitutive sequence and h is a non-erasing morphism, then h(w) is α k -substitutive for some k ∈ N. We strengthen this result in two directions. First, we show that k may be taken to be 1; hence h(w) is α k -substitutive for every k ∈ N.Second, we show that Durand's result also holds for non-erasing transductions. PreliminariesWe recall some of the main concepts that we use in the paper. For a thorough introduction to morphic sequences, automatic sequences and finite state transducers, we refer to [1,8].We are concerned with infinite sequences Σ ω over a finite alphabet Σ. We write Σ * for the set of finite words, Σ + for the finite, non-empty words, Σ ω for the infinite words, and Σ ∞ = Σ * ∪ Σ ω for all finite or infinite words over Σ.