  Frobenius problem and its many generalizations have been extensively studied
in several areas of mathematics. We study semigroups of totally positive
algebraic integers in totally real number fields, defining analogues of the
Frobenius numbers in this context. We use a geometric framework recently
introduced by Aliev, De Loera and Louveaux to produce upper bounds on these
Frobenius numbers in terms of a certain height function. We discuss some
properties of this function, relating it to absolute Weil height and obtaining
a lower bound in the spirit of Lehmer's conjecture for algebraic vectors
satisfying some special conditions. We also use a result of Borosh and Treybig
to obtain bounds on the size of representations and number of elements of
bounded height in such positive semigroups of totally real algebraic integers.
