Two classes of finite semigroups and monoids involving Lucas numbers. (Q1014272)

As is depicted in this paper, the class of finitely presented groups \(\pi_1=\langle a,b;\;a^n=b^n,\;aba^{\tfrac n2}b^{\tfrac n2}=1\rangle\) is an extension of the class of triangle groups, and also their orders are finite depending on Lucas numbers. Generally, in this paper, by considering three presentations \(\pi_1\), \(\pi_2=\langle a,b;\;a^n=b^n,\;a^2ba^{\tfrac n2}b^{\tfrac n2}=a\rangle\) and \(\pi_3=\langle a,b;\;a^n=b^n,\;a^2ba^{\tfrac n2}b^{\tfrac n2+1}=ab\rangle\), it are studied the monoid \(\text{Mon}(\pi_i)\) for \(i=1,2,3\) and the semigroup \(\text{Sem}(\pi_i)\) for \(i=2,3\), and their relationship with the group \(\text{Grp}(\pi_i)\) for \(i=1,2,3\). In detail, it is proved in this paper that 1) For all \(n\geq 2\), \(\text{Mon}(\pi_1)\) is actually a group and \(\text{Mon}(\pi_1)\cong\text{Grp}(\pi_1)\). 2) For all \(n\geq 2\), \(|\text{Sem}(\pi_2)|=|\text{Grp}(\pi_2)|+n-1\). 3) For all \(n\geq 2\), \(|\text{Sem}(\pi_3)|=|\text{Grp}(\pi_3)|+n^2\).