Topological rewriting systems applied to standard bases and - PowerPoint PPT Presentation

Topological rewriting systems applied to standard bases and syntactic algebras Cyrille Chenavier Computer Algebra Seminar - RISC Hagenberg, November 5, 2020 1 / 34

I. Motivations ⊲ Confluence property, polynomial reduction and Gröbner bases ⊲ Rewriting formal power series and standard bases II. Topological rewriting systems ⊲ Topological confluence property ⊲ Standard bases and topological confluence III. Reduction operators ⊲ Lattice structure ⊲ Lattice characterisation of topological confluence IV. Duality and syntactic algebras ⊲ Syntactic algebras ⊲ A duality criterion V. Conclusion and perspectives 2 / 34

I. MOTIVATIONS 3 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xy x − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 x yx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations Induces (under good hypotheses) ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Algebraic structures presented by oriented relations Some algorithmic Constructive methods problems in algebra in algebra • solve decision problems • compute set of representatives Classical ( e.g. , membership problem) for congruence classes • compute homological invariants • construct free resolutions of techniques ( e.g. , Tor, Ext groups) modules • analysis of functional systems • elimination theory for systems ( e.g. , integrability conditions) of equations Induces (under good hypotheses) ALGEBRAIC REWRITING Approach: orientation of relations in a structure notion of normal form ➔ example: chosen orientation in K [ x , y ] induced by yx → xy ➔ NF computation: 3 yx x + xyx − xy → 4 xyx − xy → 4 x xy − xy Remark on the case K [ x , y ] : NF monomials x n y m form a linear basis 4 / 34

Motivations Normal forms and linear bases of algebras MOTIVATING PROBLEM Given an algebra A := K � X | R � presented by generators X and relations R � K [ x , y ] = K � x , y | yx − xy � � A := K � X � / I ( R ) e . g ., Question: given an orientation of R ( e.g. , yx → xy ) do NF monomials form a linear basis of A? 5 / 34

Motivations Normal forms and linear bases of algebras MOTIVATING PROBLEM Given an algebra A := K � X | R � presented by generators X and relations R � K [ x , y ] = K � x , y | yx − xy � � A := K � X � / I ( R ) e . g ., Question: given an orientation of R ( e.g. , yx → xy ) do NF monomials form a linear basis of A? 5 / 34