Catenary degree in numerical. monoids (An application to numerical monoids generated by arithmetic sequences) D. Llena Carrasco ´ Area de Geometr´ ıa y Topolog´ ıa Universidad de Almer´ ıa Porto-March 2008
Catenary degree Credits This is a joint work with Introduction Definitions Catenary degree ◮ S. T. Chapman, (Trinity) Computing the catenary degree ◮ P . A. Garc´ ıa-S´ anchez (Granada) of the monoid example example Characterizations of the catenary degree To appear in Numerical Semigroups Generated By Forum Mathematicum Arithmetic Sequences 2 / 16
Catenary degree Set of factorizations ◮ Let S = � n 1 , . . . , n p � be a numerical monoid minimally generated by n 1 < n 2 < . . . < n p . ◮ For every s ∈ S , we define the set of factorizations of s Credits Introduction Z( s ) = { ( a 1 , . . . a p ) ∈ N p : a 1 n 1 + . . . + a p n p = s } Definitions Catenary degree Computing the catenary degree of the monoid example example Characterizations of the catenary degree Numerical Semigroups Generated By Arithmetic Sequences 3 / 16
Catenary degree Set of factorizations ◮ Let S = � n 1 , . . . , n p � be a numerical monoid minimally generated by n 1 < n 2 < . . . < n p . ◮ For every s ∈ S , we define the set of factorizations of s Credits Introduction Z( s ) = { ( a 1 , . . . a p ) ∈ N p : a 1 n 1 + . . . + a p n p = s } Definitions Catenary degree Computing the catenary degree of the monoid Support example example Let a = ( a 1 , . . . a p ) ∈ Z p We define the support of a as: Characterizations of the catenary degree Numerical supp( a ) = { i : a i � 0 } Semigroups Generated By Arithmetic Sequences 3 / 16
Catenary degree Set of factorizations ◮ Let S = � n 1 , . . . , n p � be a numerical monoid minimally generated by n 1 < n 2 < . . . < n p . ◮ For every s ∈ S , we define the set of factorizations of s Credits Introduction Z( s ) = { ( a 1 , . . . a p ) ∈ N p : a 1 n 1 + . . . + a p n p = s } Definitions Catenary degree Computing the catenary degree of the monoid Distance example example ◮ For z = ( z 1 , . . . , z p ) ∈ Z p we define its norm: Characterizations of the catenary degree Numerical Semigroups � � | z | = max { z i , | z i |} Generated By Arithmetic z i > 0 z i < 0 Sequences The length of a factorization a ∈ N p is | a | . ◮ For a , b ∈ N p we define the distance between a and b as d ( a , b ) = | a − b | Distance is the main tool in this work. The catenary degree tries to control the distance between all different factorizations of the elements of S . 3 / 16
Catenary degree Catenary degree Credits Given s ∈ S and a , b ∈ Z( s ) , then an N -chain of factorizations from a Introduction to b is a sequence a = a 1 , . . . , a t = b , a i ∈ Z( s ) such that Definitions Catenary degree d( a i , a i + 1 ) ≤ N for all i . Computing the catenary degree of the monoid example example Characterizations of the catenary degree Numerical Semigroups Generated By Arithmetic Sequences 4 / 16
Catenary degree Catenary degree Credits Given s ∈ S and a , b ∈ Z( s ) , then an N -chain of factorizations from a Introduction to b is a sequence a = a 1 , . . . , a t = b , a i ∈ Z( s ) such that Definitions Catenary degree d( a i , a i + 1 ) ≤ N for all i . Computing the catenary degree of the monoid example example Characterizations of The catenary degree of s , c( s ) , is the minimal N ∈ N ∪ {∞} such that the catenary degree for any two factorizations a , b ∈ Z( s ) , there is an N -chain from a to b . Numerical Semigroups Generated By Arithmetic Sequences 4 / 16
Catenary degree Catenary degree Credits Given s ∈ S and a , b ∈ Z( s ) , then an N -chain of factorizations from a Introduction to b is a sequence a = a 1 , . . . , a t = b , a i ∈ Z( s ) such that Definitions Catenary degree d( a i , a i + 1 ) ≤ N for all i . Computing the catenary degree of the monoid example example Characterizations of The catenary degree of s , c( s ) , is the minimal N ∈ N ∪ {∞} such that the catenary degree for any two factorizations a , b ∈ Z( s ) , there is an N -chain from a to b . Numerical Semigroups Generated By Arithmetic Sequences The catenary degree of S is c( S ) = sup { c ( s ): s ∈ S } . 4 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example example Characterizations of the catenary degree Numerical Semigroups Generated By Arithmetic Sequences 5 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example ( 0 , 1 , 3 ) ( 1 , 4 , 0 ) example Characterizations of � � ������� the catenary degree � � � � � Numerical ( 4 , 2 , 0 ) ( 7 , 0 , 0 ) Semigroups Generated By Arithmetic Sequences 5 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example 4 example ( 0 , 1 , 3 ) ( 1 , 4 , 0 ) Characterizations of 7 � the catenary degree � ������� � 5 � 6 � � � Numerical 3 Semigroups ( 4 , 2 , 0 ) ( 7 , 0 , 0 ) Generated By 3 Arithmetic Sequences 5 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example 4 example ( 0 , 1 , 3 ) ( 1 , 4 , 0 ) Characterizations of the catenary degree ������� 5 6 Numerical 3 Semigroups ( 4 , 2 , 0 ) ( 7 , 0 , 0 ) Generated By 3 Arithmetic Sequences 5 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example 4 example ( 0 , 1 , 3 ) ( 1 , 4 , 0 ) Characterizations of the catenary degree ������� 5 Numerical 3 Semigroups ( 4 , 2 , 0 ) ( 7 , 0 , 0 ) Generated By 3 Arithmetic Sequences 5 / 16
Catenary degree The catenary degree of an element Credits Let S = � 6 , 9 , 11 � Introduction Definitions Catenary degree Z ( 42 ) = { ( 0 , 1 , 3 ) , ( 1 , 4 , 0 ) , ( 4 , 2 , 0 ) , ( 7 , 0 , 0 ) } Computing the catenary degree of the monoid example 4 example ( 0 , 1 , 3 ) ( 1 , 4 , 0 ) Characterizations of the catenary degree ������� Numerical 3 Semigroups ( 4 , 2 , 0 ) ( 7 , 0 , 0 ) Generated By 3 Arithmetic So Sequences C ( 42 ) = 4 5 / 16
Catenary degree The Tool Credits R -classes Introduction Definitions Let s ∈ S and a , b ∈ Z( s ) be two factorizations of s . We say that both Catenary degree factorizations are related a R b if there exists a sequence Computing the catenary degree a = a 1 , . . . , a t = b , a i ∈ Z( s ) such that supp ( a i ) ∩ supp ( a i + 1 ) � ∅ . of the monoid example example Characterizations of the catenary degree Numerical Semigroups Generated By Arithmetic Sequences 6 / 16
Catenary degree The Tool Credits R -classes Introduction Definitions Let s ∈ S and a , b ∈ Z( s ) be two factorizations of s . We say that both Catenary degree factorizations are related a R b if there exists a sequence Computing the catenary degree a = a 1 , . . . , a t = b , a i ∈ Z( s ) such that supp ( a i ) ∩ supp ( a i + 1 ) � ∅ . of the monoid example example Characterizations of Proposition the catenary degree Numerical Semigroups The distance between elements a and b in different R -clases is Generated By Arithmetic d ( a , b ) = max {| a | , | b |} . Sequences We can control the elements with more than one R -class. 6 / 16
Catenary degree Ap´ ery set Credits ◮ For every n i we define the Ap´ ery set of S respect to n i as Introduction Definitions Catenary degree Ap ( S , n i ) = { s ∈ S : s − n i � S } Computing the catenary degree of the monoid ◮ Ap ( S , n i ) = { w ( 0 ) , w ( 1 ) , . . . , w ( n i − 1 ) } with w ( j ) the least element example example in S congruent with j modulo n i Characterizations of the catenary degree Numerical Semigroups Generated By Arithmetic Sequences 7 / 16
Catenary degree Ap´ ery set Credits ◮ For every n i we define the Ap´ ery set of S respect to n i as Introduction Definitions Catenary degree Ap ( S , n i ) = { s ∈ S : s − n i � S } Computing the catenary degree of the monoid ◮ Ap ( S , n i ) = { w ( 0 ) , w ( 1 ) , . . . , w ( n i − 1 ) } with w ( j ) the least element example example in S congruent with j modulo n i Characterizations of the catenary degree Numerical Semigroups Generated By Important Result Arithmetic Sequences The elements of S with more than one R -class are of the form w + n j where w ∈ Ap ( S , n 1 ) and j = 2 , . . . , p . 7 / 16
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