A graded monomial ordering that compares total degree first and, for equal total degree, uses reverse lexicographic tie-breaking: compare exponent vectors from the last variable toward the first and, at the rightmost index where they differ, the monomial with the smaller exponent at that index is greater. It is widely used for Gröbner-basis computations because of its favorable practical behavior.

Principle

Primary key: total degree. Secondary key: reverse lexicographic comparison of exponent vectors, where the rightmost differing exponent determines the order with the smaller exponent making the monomial larger.

Demonstration

With x>y>z, compare x^2y (2,1,0) and xy^2 (1,2,0); both have total degree 3. Looking from z to x, the first difference is at y (1 vs. 2). Since 1<2, x^2y > xy^2 under graded reverse lexicographic (Grevlex) order.

Misapplication

Assuming Grevlex enforces the same elimination patterns as lex or that it preserves coefficient sparsity in all cases; Grevlex optimizes degree behavior but does not necessarily yield elimination ideals by simple projection the way lex can.

Consequence

Grevlex often produces Gröbner bases with lower degree growth and better computational efficiency in practice; it is a common default in computer algebra systems for these reasons.

Reversal

Switch to GrLex or pure lex: prioritizing lexicographic tie-breaks or removing degree-first priority will change which monomials are leading and can make elimination properties explicit at the cost of degree behavior.

Boundary

Defined for ordinary polynomial rings with total-degree grading; it does not automatically cover weighted reverse-lex orders or noncommutative settings without adaptation.

Semantic Tension

Competes with GrLex and lex: Grevlex balances degree-first priorities with a reverse tie-break that tends to favor sparser high-degree monomials, whereas lex enforces strict variable precedence and GrLex uses forward lex tie-breaks.

Synthesis

Grevlex orders monomials by total degree and resolves ties by reverse lexicographic comparison of exponents from last to first, trading strict variable precedence for favorable degree-growth properties and often superior practical performance.