TY - JOUR

T1 - MULTIPLICATION TABLES AND WORD-HYPERBOLICITY IN FREE PRODUCTS OF SEMIGROUPS, MONOIDS AND GROUPS

AU - Nyberg-Brodda, Carl-Fredrik

N1 - Cited by: 0; All Open Access, Green Open Access, Hybrid Gold Open Access

PY - 2023/3/17

Y1 - 2023/3/17

N2 - This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman ['Word hyperbolic semigroups', Math. Proc. Cambridge Philos. Soc. 136(3) (2004), 513-524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a -multiplication table, where is any reversal-closed super-. In particular, we deduce that the free product of two groups with with respect to indexed multiplication tables again has an with respect to an indexed multiplication table. The © 2023 Author(s).

AB - This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman ['Word hyperbolic semigroups', Math. Proc. Cambridge Philos. Soc. 136(3) (2004), 513-524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a -multiplication table, where is any reversal-closed super-. In particular, we deduce that the free product of two groups with with respect to indexed multiplication tables again has an with respect to an indexed multiplication table. The © 2023 Author(s).

U2 - 10.1017/S1446788723000010

DO - 10.1017/S1446788723000010

M3 - Article

SN - 1446-8107

VL - 85

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 10-12

ER -