Cyclic rational semirings A comparison between the factorization invariants of numerical monoids and Puiseux monoids Marly Gotti March 22, 2019 AMS Sectional Meeting, University of Hawaii at Manoa
Table of contents 1. General comparison between numerical and Puiseux monoids 2. Defjnitions and background 3. Introduction to cyclic rational semirings 4. Set of lengths, delta set, and catenary degree 5. Elasticity 6. Comparing two classes of nicely generated monoids 1
General comparison between numerical and Puiseux monoids
Numerical and Puiseux monoids Defjnition numerical monoid. Defjnition Note : Every numerical monoid is a Puiseux monoid. 2 If N is a submonoid of ( N 0 , +) such that N 0 \ N is fjnite, then N is called a A submonoid of ( Q ≥ 0 , +) is called a Puiseux monoid.
Numerical and Puiseux monoids Defjnition numerical monoid. Defjnition Note : Every numerical monoid is a Puiseux monoid. 2 If N is a submonoid of ( N 0 , +) such that N 0 \ N is fjnite, then N is called a A submonoid of ( Q ≥ 0 , +) is called a Puiseux monoid.
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids There exists a Puiseux No known general results. Catenary degree tic if and only if N is isomorphic to accepted. Moreover, N is fully elas- Let N be a numerical monoid Elasticity 3 behavior. System of sets of lengths there is a numerical monoid N and Sets of lengths are arithmetic mul- Let M be a Puiseux monoid Sets of lengths can have arbitrary tiprogressions. Also, for L ⊆ N ≥ 2 , monoid M such that for any L ⊆ x ∈ N with L ( x ) = L . N ≥ 2 , there is an x ∈ M with L ( x ) = L . ρ ( N ) = max A ( N ) If M is atomic, then ρ ( M ) = ∞ if 0 min A ( N ) is always fjnite and is a limit point of A ( M ) and ρ ( M ) = sup A ( M ) inf A ( M ) otherwise. Moreover, ρ ( M ) is ( N 0 , +) . accepted if and only if A ( M ) has a min and a max in Q . c ( N ) ≤ F ( N )+max A ( N ) + 1. min A ( N )
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids There exists a Puiseux No known general results. Catenary degree tic if and only if N is isomorphic to accepted. Moreover, N is fully elas- Let N be a numerical monoid Elasticity 3 behavior. System of sets of lengths there is a numerical monoid N and Sets of lengths are arithmetic mul- Let M be a Puiseux monoid Sets of lengths can have arbitrary tiprogressions. Also, for L ⊆ N ≥ 2 , monoid M such that for any L ⊆ x ∈ N with L ( x ) = L . N ≥ 2 , there is an x ∈ M with L ( x ) = L . ρ ( N ) = max A ( N ) If M is atomic, then ρ ( M ) = ∞ if 0 min A ( N ) is always fjnite and is a limit point of A ( M ) and ρ ( M ) = sup A ( M ) inf A ( M ) otherwise. Moreover, ρ ( M ) is ( N 0 , +) . accepted if and only if A ( M ) has a min and a max in Q . c ( N ) ≤ F ( N )+max A ( N ) + 1. min A ( N )
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids There exists a Puiseux No known general results. Catenary degree tic if and only if N is isomorphic to accepted. Moreover, N is fully elas- Let N be a numerical monoid Elasticity 3 behavior. System of sets of lengths there is a numerical monoid N and Sets of lengths are arithmetic mul- Let M be a Puiseux monoid Sets of lengths can have arbitrary tiprogressions. Also, for L ⊆ N ≥ 2 , monoid M such that for any L ⊆ x ∈ N with L ( x ) = L . N ≥ 2 , there is an x ∈ M with L ( x ) = L . ρ ( N ) = max A ( N ) If M is atomic, then ρ ( M ) = ∞ if 0 min A ( N ) is always fjnite and is a limit point of A ( M ) and ρ ( M ) = sup A ( M ) inf A ( M ) otherwise. Moreover, ρ ( M ) is ( N 0 , +) . accepted if and only if A ( M ) has a min and a max in Q . c ( N ) ≤ F ( N )+max A ( N ) + 1. min A ( N )
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids There exists a Puiseux No known general results. Catenary degree tic if and only if N is isomorphic to accepted. Moreover, N is fully elas- Let N be a numerical monoid Elasticity 3 behavior. System of sets of lengths there is a numerical monoid N and Sets of lengths are arithmetic mul- Let M be a Puiseux monoid Sets of lengths can have arbitrary tiprogressions. Also, for L ⊆ N ≥ 2 , monoid M such that for any L ⊆ x ∈ N with L ( x ) = L . N ≥ 2 , there is an x ∈ M with L ( x ) = L . ρ ( N ) = max A ( N ) If M is atomic, then ρ ( M ) = ∞ if 0 min A ( N ) is always fjnite and is a limit point of A ( M ) and ρ ( M ) = sup A ( M ) inf A ( M ) otherwise. Moreover, ρ ( M ) is ( N 0 , +) . accepted if and only if A ( M ) has a min and a max in Q . c ( N ) ≤ F ( N )+max A ( N ) + 1. min A ( N )
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids atomic if 0 is not a limit point of M . M can be a BFM and not an FFM. Always. Is it an FF-monoid (FFM)? M can be atomic and not a BFM. Always. Is it a BF-monoid (BFM)? Always. Let N be a numerical monoid Is it atomic? monoid. if M is isomorphic to a numerical M is fjnitely generated if and only Always. Is it fjnitely generated? Let M be a Puiseux monoid 4 � 1 / 2 n | n ∈ N 0 � is not atomic. M is
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids atomic if 0 is not a limit point of M . M can be a BFM and not an FFM. Always. Is it an FF-monoid (FFM)? M can be atomic and not a BFM. Always. Is it a BF-monoid (BFM)? Always. Let N be a numerical monoid Is it atomic? monoid. if M is isomorphic to a numerical M is fjnitely generated if and only Always. Is it fjnitely generated? Let M be a Puiseux monoid 4 � 1 / 2 n | n ∈ N 0 � is not atomic. M is
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids atomic if 0 is not a limit point of M . M can be a BFM and not an FFM. Always. Is it an FF-monoid (FFM)? M can be atomic and not a BFM. Always. Is it a BF-monoid (BFM)? Always. Let N be a numerical monoid Is it atomic? monoid. if M is isomorphic to a numerical M is fjnitely generated if and only Always. Is it fjnitely generated? Let M be a Puiseux monoid 4 � 1 / 2 n | n ∈ N 0 � is not atomic. M is
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids atomic if 0 is not a limit point of M . M can be a BFM and not an FFM. Always. Is it an FF-monoid (FFM)? M can be atomic and not a BFM. Always. Is it a BF-monoid (BFM)? Always. Let N be a numerical monoid Is it atomic? monoid. if M is isomorphic to a numerical M is fjnitely generated if and only Always. Is it fjnitely generated? Let M be a Puiseux monoid 4 � 1 / 2 n | n ∈ N 0 � is not atomic. M is
Table 1: Factorization invariants of numerical monoids vs. Puiseux monoids atomic if 0 is not a limit point of M . M can be a BFM and not an FFM. Always. Is it an FF-monoid (FFM)? M can be atomic and not a BFM. Always. Is it a BF-monoid (BFM)? Always. Let N be a numerical monoid Is it atomic? monoid. if M is isomorphic to a numerical M is fjnitely generated if and only Always. Is it fjnitely generated? Let M be a Puiseux monoid 4 � 1 / 2 n | n ∈ N 0 � is not atomic. M is
Defjnitions and background
Defjnitions Let M be a reduced commutative cancellative monoid. fjnitely generated if it can be generated by a fjnite set. atomic. 5 • M • denotes the set M \{ 0 } . • We write M = � S � when M is generated by a set S . We say that M is • An element a ∈ M • is called an atom provided that for each pair of elements x , y ∈ M such that a = x + y either x = 0 or y = 0. The set of atoms of M is denoted by A ( M ) . If A ( M ) generates M , then M is called • The factorization monoid of M is the free commutative monoid on A ( M ) and is denoted by Z ( M ) . The elements of Z ( M ) are called factorizations. • If z = a 1 + · · · + a n is a factorization of M for some a 1 , . . . , a n ∈ A ( M ) , then n is called the length of z and is denoted by | z | .
Defjnitions Let M be a reduced commutative cancellative monoid. fjnitely generated if it can be generated by a fjnite set. atomic. 5 • M • denotes the set M \{ 0 } . • We write M = � S � when M is generated by a set S . We say that M is • An element a ∈ M • is called an atom provided that for each pair of elements x , y ∈ M such that a = x + y either x = 0 or y = 0. The set of atoms of M is denoted by A ( M ) . If A ( M ) generates M , then M is called • The factorization monoid of M is the free commutative monoid on A ( M ) and is denoted by Z ( M ) . The elements of Z ( M ) are called factorizations. • If z = a 1 + · · · + a n is a factorization of M for some a 1 , . . . , a n ∈ A ( M ) , then n is called the length of z and is denoted by | z | .
Defjnitions Let M be a reduced commutative cancellative monoid. fjnitely generated if it can be generated by a fjnite set. atomic. 5 • M • denotes the set M \{ 0 } . • We write M = � S � when M is generated by a set S . We say that M is • An element a ∈ M • is called an atom provided that for each pair of elements x , y ∈ M such that a = x + y either x = 0 or y = 0. The set of atoms of M is denoted by A ( M ) . If A ( M ) generates M , then M is called • The factorization monoid of M is the free commutative monoid on A ( M ) and is denoted by Z ( M ) . The elements of Z ( M ) are called factorizations. • If z = a 1 + · · · + a n is a factorization of M for some a 1 , . . . , a n ∈ A ( M ) , then n is called the length of z and is denoted by | z | .
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