The finite basis problem for monoids of triangular boolean matrices

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 $. 1 The finite basis problem for matrix monoids An algebra $A $ is said to be finitely based if all identities holding in $A $ follow from a finite set of such identities (an identity basis of $A $); otherwise $A $ is called nonfinitely based. While every finite group is finitely based (Oates and Powell [5] $) $ , among finite monoids there exist nonfinitely based ones. The first example of a nonfinitely based finite monoid (due to Perkins [6]) was the Brandt monoid $B_{2}^{1} $ formed by the six matrice