We provide a general framework for computing mixing times of finite Markov chains whose semigroup's minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribution in the probabilities. From these we obtain bounds on the mixing time. In addition, we provide a new Markov chain on linear extension of a poset with $n$ vertices, inspired by but different from the promotion Markov chain of Ayyer, Klee, and the last author. The mixing time of this Markov chain is $O(n \log n)$.

We develop the rigorous notion of a model for understanding state transition systems by hierarchical coordinate systems. Using this we motivate an algebraic definition of the complexity of biological systems, comparing it to other candidates such as genome size and number of cell types. We show that our complexity measure is the unique maximal complexity measure satisfying a natural set of axioms. This reveals a strong relationship between hierarchical complexity in biological systems and the area of algebra known as global semigroup theory. We then study the rate at which hierarchical complexity can evolve in biological systems assuming evolution is "as slow as possible" from the perspective of computational power of organisms. Explicit bounds on the evolution of complexity are derived showing that, although the evolutionary changes in hierarchical complexity are bounded, in some circumstances complexity may more than double in certain "genius jumps" of evolution. In fact, examples show that our bounds are sharp. We sketch the structure where such complexity jumps are known to occur and note some similarities to previously identified mechanisms in biological evolutionary transitions. We also address the question of, How fast can complexity evolve over longer periods of time? Although complexity may more than double in a single generation, we prove that in a smooth sequence of t "inclusion" steps, complexity may grow at most from N to (N + 1)t + N, a linear function of number of generations t, while for sequences of "mapping" steps it increases by at most t. Thus, despite the fact that there are major transitions in which complexity jumps are possible, over longer periods of time, the growth of complexity may be broken into maximal intervals on which it is bounded above in the manner described.

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups. If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety. We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.