  We consider algebras over a field K defined by a presentation K <x_1,..., x_n
: R >, where $R$ consists of n choose 2 square-free relations of the form x_i
x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one
of the relations. Certain sufficient conditions for the algebra to be
noetherian and PI are determined. For this, we prove more generally that right
noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous
relations satisfy a polynomial identity. The structure of the underlying
monoid, defined by the same presentation, is described. This is used to derive
information on the prime radical and minimal prime ideals. Some examples are
described in detail. Earlier, Etingof, Schedler and Soloviev, Gateva-Ivanova
and Van den Bergh, and the authors considered special classes of such algebras
in the contexts of noetherian algebras, Grobner bases, finitely generated
solvable groups, semigroup algebras, and set theoretic solutions of the
Yang-Baxter equation.
