Steiner Triple Systems And Their Endomorphisms

The research on Steiner triple systems has been ongoing for a long time.Besides combinatorial methods,algebraic methods can also be applied.Because Steiner triple systems can be transformed into Steiner quasigroups and Steiner loops,we can use these two algebraic structures to study Steiner triple systems.Also,the automorphism groups of Steiner triple systems can be used to study properties of Steiner triple systems.Since the automorphism groups of most of the Steiner triple systems are trivial,endomorphisms of Steiner triple systems become more important.We study endomorphisms of Steiner triple systems,including endomorphisms of Steiner quasigroups and endomorphisms of Steiner loops.For Steiner quasigroups,we prove that Steiner quasigroup homomorphism images and preimages are Steiner quasigroups.We prove that all preimages are of the same order,and give an example whose two preimages are of the same order,but not isomorphic to each other.Furthermore,the weak fundamental homomorphism theorem of Steiner quasigroups is given,which gives an order division relation.We study Steiner quasigroups with only trivial endomorphisms and give two infinite families of Steiner quasigroups with only trivial endomorphisms.For Steiner loops,we prove that homomorphic images and kernels between Steiner loops are Steiner loops,and present the fundamental homomorphism theorem of Steiner loops.Then,we reveal the relationship between endomorphisms of Steiner quasigroups and endomorphisms of Steiner loops.Finally,we consider endomorphisms of Steiner triple systems with order ? 21.We investigate properties of Steiner quasigroups and Steiner loops of small order whose endomorphisms are nontrivial.For instance,we prove that every Steiner quasigroup of order 21 that contains 7 subquasigroups of order 9 and 1 subquasigroup of order 7 has a nontrivial endomorphism whose image is of order 7.