Two classes of finite semigroups and monoids involving Lucas numbers

摘要

The class of finitely presented groups a,b a~n = b~n, aba~([n/2])b~([n/2])=1 is an extension of the class of triangle groups studied recently. These groups are finite and their orders depend on the Lucas numbers. In this paper, by considering the three presentations π_1=a,b a~n = b~n, aba~([n/2])b ~([n/2])=1, π_2=a,b a~n = b~n, a ~2ba~([n/2])b~([n/2])=a and π_3=a,b a~n = b~n, a~2ba~([n/2])b ~([n/2]+1)=ab, we study Mon(π _i), i=1,2,3, and Sg(π _i), i=2,3, for their finiteness. In this investigation, we find their relationship with Gp(π _i), where Mon(π), Sg(π) and Gp(π) are used for the monoid, the semigroup and the group presented by the presentation π, respectively.