Full Monoid Research Articles

On the Interplay Between Černý and Babai’s Conjectures

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with [Formula: see text] states in this class, we prove that the reset threshold is upper-bounded by [Formula: see text] and can attain the value [Formula: see text]. In addition, we study diameters of the pair digraphs of permutation automata and construct [Formula: see text]-state permutation automata with diameter [Formula: see text].

The combinatorial game theory of well-tempered scoring games

The general behavior of combinatorial scoring games is not well-understood. In this paper, we focus on a special class of “well-tempered” scoring games. By analogy with Grossman and Siegel’s notion of even- and odd-tempered normal play games, we declare a dicot scoring game to be even-tempered if all its options are odd-tempered, and odd-tempered if it is not atomic and all its options are even-tempered. Games of either sort are called well-tempered. These show up naturally when analyzing one of the “knot games” introduced by Henrich et al. We consider disjunctive sums of well-tempered scoring games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal play partizan games. We isolate a special class of inversive well-tempered games which behave like normal play partizan games or like the “Milnor games” considered by Milnor and Hanner. In particular, inversive games (modulo equivalence) form a partially ordered abelian group, and there is an effective description of the partial order. Moreover, the full monoid of well-tempered scoring games (modulo equivalence) admits a complete description in terms of the group of inversive games. We also describe several examples of well-tempered scoring games and provide dictionaries listing the values of some small positions in two of these games.

A Note on Free Idempotent Generated Semigroups over the Full Monoid of Partial Transformations

Recently, Gray and Ru\\v skuc proved that if e is a rank k idempotent transformation of the set {1,…, n} to itself and k ≤ n − 2, then the maximal subgroup of the free idempotent generated semigroup over the full transformation monoid 𝒯 n containing e is isomorphic to the symmetric group 𝒮 k . We prove that the same holds when 𝒯 n is replaced by 𝒫𝒯 n , the full monoid of partial transformations on {1,…, n}.

FULL MONOIDS AND MAXIMAL CODES

In recent years codes that are not Uniquely Decipherable (UD) were studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a free monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal set. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code. Finally we generalize some properties of UD codes.

Monoids and Maximal Codes

In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code.