The unique factorization theorem

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

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The Unique Factorization Theorem: From Euclid to Gauss

Mathematics Magazine ◽

10.1080/0025570x.1980.11976835 ◽

1980 ◽

Vol 53 (2) ◽

pp. 96-100 ◽

Cited By ~ 1

Author(s):

Mary Joan Collison

Keyword(s):

Factorization Theorem ◽

Unique Factorization

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CONES OF CURVES AND OF LINE BUNDLES ON SURFACES ASSOCIATED WITH CURVES HAVING ONE PLACE AT INFINITY

Proceedings of the London Mathematical Society ◽

10.1112/s0024611502013394 ◽

2002 ◽

Vol 84 (3) ◽

pp. 559-580 ◽

Cited By ~ 12

Author(s):

ANTONIO CAMPILLO ◽

OLIVIER PILTANT ◽

ANA J. REGUERA-LÓPEZ

Keyword(s):

Polyhedral Cone ◽

Subject Classification ◽

Factorization Theorem ◽

Line Bundles ◽

Unique Factorization ◽

Primary 14C20 ◽

New Class ◽

Secondary 14E05 ◽

Cone Of Curves ◽

Projective Surfaces

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

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A remark on the unique factorization theorem.

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/00910143 ◽

1957 ◽

Vol 9 (1) ◽

pp. 143-145 ◽

Cited By ~ 25

Author(s):

Department of Mathematics Kyoto University ◽

Masayoshi NAGATA

Keyword(s):

Factorization Theorem ◽

Unique Factorization

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Spaces of lower semicontinuous set-valued maps II

Mathematica Slovaca ◽

10.2478/s12175-010-0031-9 ◽

2010 ◽

Vol 60 (4) ◽

Cited By ~ 1

Author(s):

R. McCoy

Keyword(s):

Lower Semicontinuous ◽

Extension Theorem ◽

Factorization Theorem ◽

Unique Factorization ◽

Previous Part

AbstractThis is a continuation of “Spaces of lower semicontinuous setvalued maps I”. Together, these two parts contain two interrelated main theorems. In the previous part I, the Extension Theorem is proved, which says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). In this part II, the Factorization Theorem is proved, which says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.

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