Indispensable monomials of toric ideals and Markov bases

Extending the notion of indispensable binomials of a toric ideal [Takemura, Akimichi, Aoki, Satoshi, 2004. Some characterizations of minimal Markov basis for sampling from discrete conditional distributions. Ann. Inst. Statist. Math. 56 (1), 1–17; Ohsugi, Hidefumi, Hibi, Takayuki, 2005. Indispensable binomials of finite graphs. J. Algebra Appl. 4 (4), 421–434], we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomial generators of a toric ideal. Some examples of indispensable monomials from statistical models for contingency tables are given.