A monomial order that first compares total degree (sum of exponents) and, when total degrees coincide, breaks ties lexicographically according to a chosen variable ordering. It balances degree-based prioritization with a deterministic lexicographic tie-break.

Principle

Primary key: total degree. Secondary key: lexicographic comparison of exponent vectors using a fixed variable precedence. This two-level key makes the order graded with lexicographic resolution of ties.

Demonstration

With x>y>z, compare x^2y (degree 3) and xy^3 (degree 4). Since 3<4, xy^3 > x^2y under Graded Lexicographic (GrLex). If degrees match, fall back to lex comparison of exponents.

Misapplication

Treating GrLex as equivalent to pure lex for elimination: because GrLex prioritizes total degree, it does not guarantee the same elimination properties as lex ordering and may fail to produce elimination ideals in the same straightforward way.

Consequence

When used appropriately, GrLex reduces uncontrolled degree explosion compared with pure lex in many Gröbner-basis computations, while preserving a fixed, reproducible tie-break via lexicographic comparison.

Reversal

Reverse the priority by using lex (no degree priority) or use a graded reverse lexicographic (Grevlex) tie-breaker instead; such reversals change which monomials are selected as leaders and affect algorithmic complexity.

Boundary

Defined for commutative polynomial rings with simple total-degree grading; it does not subsume weighted graded orders or block orders without explicit modification.

Semantic Tension

Competes with Grevlex: both are graded (degree-first) orders, but their tie-breakers (lex vs. reverse-lex) yield different Gröbner bases and different practical performance characteristics.

Synthesis

Graded lexicographic order orders monomials by total degree and resolves equal-degree ties by lexicographic comparison of exponent vectors, trading strict variable precedence for degree-awareness and reproducible tie-breaking.