We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model${\text{IPM}}_k$(n), a generalization of the sand pile model$\text{SPM}$(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice ${\text{IPM}}_k$(n): this lets us design an algorithm which generates all the ice piles of ${\text{IPM}}_k$(n) in amortized time O(1) and in space O($\sqrt{n}$).

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model${\text{IPM}}_k$(n), a generalization of the sand pile model$\text{SPM}$(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice ${\text{IPM}}_k$(n): this lets us design an algorithm which generates all the ice piles of ${\text{IPM}}_k$(n) in amortized time O(1) and in space O($\sqrt{n}$).

A $\Sigma$-labeled $n$-poset is an (at most) countable set, labeled in the set $\Sigma$, equipped with $n$ partial orders. The collection of all $\Sigma$-labeled $n$-posets is naturally equipped with $n$ binary product operations and $n$$\omega$-ary product operations. Moreover, the $\omega$-ary product operations give rise to $n$$...$

A Σ-labeled n-poset is an (at most) countable set, labeled in the set Σ, equipped with n partial orders. The collection of all Σ-labeled n-posets is naturally equipped with n binary product operations and nω-ary product operations. Moreover, the ω-ary product operations give rise to nω-power operations. We show that those Σ-labeled n-posets that can be generated from the singletons by the binary and ω-ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection...