Lattice-ordered triangular matrix algebras (Q837022)

The set \(T^0_n\) of all \((n\times n)\)-matrices \(E_{ij}\) defined for \(1\leq i\leq j\leq n\) such that the \(ij\)-entry is equal to \(1\) and all other entries are 0, including the zero matrix, forms a semigroup with respect to the usual matrix multiplication, the semigroup of triangular matrix units. In this paper, first necessary and sufficient conditions for a semigroup with zero to be isomorphic to a semigroup of triangular matrix units are given. More generally, for any group \(G\) the semigroup \(T^0_n[G]= \{(i,a,j)\mid 1\leq i\leq j\leq n,a\in G\}\cup\{0\}\) with respect to the operation \((i,a,j)(r,b,s)= (i,ab,s)\) if \(j=r\) and \(=0\) if \(j\neq r\), with \(0\) as the zero element, is considered (this is not the Rees-matrix semigroup over \(G\), since the elements \((i,a,j)\) with \(i> j\) are missing). Again, conditions for a semigroup with zero to be isomorphic with such a semigroup \(T^0_n[G]\) are given. Finally, the semigroup \(\ell\)-algebra \(F[S]\) of a semigioup \(S\) with zero over a totally ordered field \(F\) is studied. The main result provides necessary and sufficient conditions for an \(\ell\)-unital semigroup \(\ell\)-algebra \(L\) over \(F\) to be isomorphic with \(T_n(F)\), \(n\geq 2\), the \((n\times n)\)-triangular matrix algebra over \(F\) endowed with the entrywise lattice order.