Two semigroups are said to be distinct if they are neither isomorphic nor anti-isomorphic. Although there exist 1373 distinct monoids of order six, only two are known to be non-finitely based. In the present dissertation, the finite basis property of the other 1371 distinct monoids of order six is verified. Since it is long established that all semigroups of order five or less are finitely based, the two known non-finitely based monoids of order six are the only examples of minimal order.Acknowledgements. The authors would like to thank the following people: the anonymous reviewer for his comments and suggestions, Avraham Trahtman for information on the finite basis problem for semigroups of order five, Helmut Jürgensen for providing a copy of all multiplication tables of semigroups of order up to six [13], Eric Postpischil for assistance in generating the multiplication tables of monoids in Appendix A, Ken W. K. Lee for testing these monoids against Conditions 1-9 in Chapter 3 with Java, and James D. Mitchell for independently performing the same test with GAP [7].

No abstract

We show that the monoid of all $n\cross n$ upper triangular boolean matrices has no finite identity basis whenever $n>3$ . The identities of its submonoid consisting of matrices in which all diagonal entries are 1 possess a finite basis if and only if $n\leq 4$ .

No abstract

Let T n (F) denote the monoid of all upper triangular n × n matrices over a finite field F. It has been shown by Volkov and Goldberg that T n (F) is nonfinitely based if |F| > 2 and n ≥ 4, but the cases when |F| > 2 and n = 2, 3 or when |F| = 2 have remained open. In this paper, it is shown that the monoid T 2 (F) is finitely based when |F| = 2, and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by T 2 (F) with |F| = 2 are determined.2010 Mathematics subject classification: primary 20M07.