Greens Relations in Finite Transformation Semigroups

We consider the complexity of Green’s relations when the semigroup is given by transformations on a finite set. Green’s relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.

💡 Research Summary

The paper investigates the structural complexity of Green’s relations (R, L, J) in finite transformation semigroups, i.e., semigroups generated by functions on a finite set Q of size n. Green’s relations are interpreted as reachability in the right, left, or two‑sided Cayley graph of the semigroup; the equivalence classes correspond to strongly connected components. The authors first observe that the number of J‑classes (and consequently R‑ and L‑classes) can be as large as Θ(nⁿ), matching the trivial upper bound given by the size of the semigroup itself. This is already known for transformation semigroups generated by three maps, which can achieve the full nⁿ size.

The main focus is on the “height” of these relations, defined as the maximal length ℓ of a strict chain s₁ >ᴿ … >ᴿ s_ℓ (similarly for L and J). While the naïve upper bound is again nⁿ, the authors prove a much stronger lower bound for the R‑height: it grows exponentially, specifically 2^{Θ(n)}. For L‑ and J‑height they obtain a weaker Ω(2ⁿ) bound.

To obtain these results the paper introduces a novel combinatorial model called a token machine. A token machine consists of a finite set of cells C and a set I of partial transformations (instructions) on C. A computation is a sequence of configurations R₀ → R₁ → … → R_ℓ obtained by applying a word over I, where each configuration has the same cardinality and the transition is defined by the partial map. A computation is progressing if all intermediate configurations are distinct and any unused instruction would strictly reduce the size of the configuration; it is maximal if no instruction can be applied without reducing the size at the final configuration. Proposition 10 shows that a maximal progressing computation of length ℓ yields an R‑chain of length ℓ in the semigroup generated by I, thus providing a lower bound on the R‑height.

The construction proceeds in two stages. First, for an arbitrary generating set Σ, the authors give a simple augmentation that adds three new states and one new generator, producing a transformation semigroup with at least |T| distinct J‑classes (Proposition 7). This shows that the class‑count bound is tight up to a constant factor.

The second, more intricate stage deals with a fixed alphabet. Using the token‑machine framework, they design a system with n cells (half “tokens”, half “holes”) and a constant set of instructions (size five). The instructions rotate tokens, move them into holes, and perform controlled swaps, ensuring that each step of a computation preserves the number of tokens while strictly progressing toward a terminal configuration. Lemma 8 establishes that if Q·x = Q·xy then x R xy, and Proposition 9 bounds the R‑height of any partial‑transformation semigroup on n states by 2ⁿ. By carefully arranging the instructions, they construct a maximal progressing computation of length Ω(2ⁿ / n^{9.5}), thereby proving Theorem 2: for a sequence of minimal deterministic finite automata (Aₙ) over a fixed alphabet, the transition semigroup T(Aₙ) has R‑height in Ω(2ⁿ / n^{9.5}) (and analogous bounds for L‑ and J‑height).

Finally, the authors translate these semigroup results to automata theory. For each n there exists a minimal DFA with n states over an alphabet of size five whose transition semigroup has at least (n−4)·n−4 J‑classes (Theorem 1). Moreover, the same family of automata exhibits the exponential R‑height described above. Since the syntactic semigroup of a regular language coincides with the transition semigroup of its minimal DFA, the bounds also apply to syntactic semigroups.

In summary, the paper provides (i) tight asymptotic bounds on the number of Green’s equivalence classes in finite transformation semigroups, (ii) an exponential lower bound on the height of R‑chains (and comparable bounds for L and J) even when the generating alphabet is fixed, and (iii) a concrete method—via token machines—to construct transformation semigroups (and corresponding minimal automata) that realize these bounds. These contributions deepen the understanding of the internal complexity of transformation semigroups and bridge it to automata theory, offering new tools for analyzing the algebraic structure of regular languages.