  To determine and analyze arbitrary left non-degenerate set-theoretic
solutions of the Yang-Baxter equation (not necessarily bijective), we introduce
an associative algebraic structure, called a YB-semitruss, that forms a
subclass of the category of semitrusses as introduced by Brzezi\'nski.
Fundamental examples of YB-semitrusses are structure monoids of left
non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova
and Van den Bergh introduced structure monoids and showed their importance (as
well as that of the structure algebra) for studying involutive non-degenerate
solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to
deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield
a unified treatment of these different algebraic structures. The algebraic
structure of YB-semitrusses is investigated, and as a consequence, it is
proven, for example, that any finite left non-degenerate set-theoretic solution
of the Yang-Baxter equation is right non-degenerate if and only if it is
bijective. Furthermore, it is shown that some finite left non-degenerate
solutions can be reduced to non-degenerate solutions of smaller size. The
structure algebra of a finitely generated YB-semitruss is an algebra defined by
homogeneous quadratic relations. We prove that it often is a left Noetherian
algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial
identity, but in general, it is not right Noetherian.
