scholarly journals Backtracking with cut via a distributive law and left-zero monoids

2017 ◽  
Vol 27 ◽  
Author(s):  
MACIEJ PIRÓG ◽  
SAM STATON
Keyword(s):  
Left Zero ◽  
Free Left ◽  
Distributive Law

AbstractWe employ the framework of algebraic effects to augment the list monad with the pruning cut operator known from Prolog. We give two descriptions of the resulting monad: as the monad of free left-zero monoids, and as a composition via a distributive law of the list monad and the ‘unary idempotent operation’ monad. The scope delimiter of cut arises as a handler.

A NORMAL FORM FOR THE FREE LEFT DISTRIBUTIVE LAW

1994 ◽  
Vol 04 (04) ◽  
pp. 499-528 ◽  
Author(s):  
PATRICK DEHORNOY
Keyword(s):  
Normal Form ◽  
New Normal ◽  
Free Left ◽  
Distributive Law ◽  
Primitive Recursive ◽  
Made In

We construct a new normal form for one variable terms up to left distributivity. The proof that this normal form exists for every term is considerably simpler than the corresponding proof for the forms previously introduced by Richard Laver. In particular the determination of the present normal form can be made in a primitive recursive way.

The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Richard Laver ◽  
Sheila Miller
Keyword(s):  
Normal Form ◽  
Set Theory ◽  
Large Cardinal ◽  
Braid Groups ◽  
Division Algorithm ◽  
Free Left ◽  
Distributive Law ◽  
The Law

AbstractThe left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity of the iterated left division ordering <L of A, the connections of A to the braid groups, and an extension P of A obtained by freely adding a composition operation. This is followed by a simplified proof of the division algorithm for P, which produces a normal form for terms in A and is a powerful tool in the study of A.

Richard Laver. The left distributive law and the freeness of an algebra of elementary embeddings. Advances in mathematics, vol. 91 (1992), pp. 209–231. - Richard Laver. A division algorithm for the free left distributive algebra. Logic Colloquium '90, ASL summer meeting in Helsinki, edited by J. Oikkonen and J. Väänänen, Lecture notes in logic, no. 2, Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, pp. 155–162. - Richard Laver. On the algebra of elementary embeddings of a rank into itself. Advances in mathematics, vol. 110 (1995), pp. 334–346. - Richard Laver. Braid group actions on left distributive structures, and well orderings in the braid groups. Journal of pure and applied algebra, vol. 108 (1996), pp. 81–98. - Patrick Dehornoy. An alternative proof of Laver's results on the algebra generated by an elementary embedding. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematics Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 27–33. - Patrick Dehornoy. Braid groups and left distributive operations. Transactions of the American Mathematical Society, vol. 345 (1994), pp. 115–150. - Patrick Dehornoy. A normal form for the free left distributive law. International journal of algebra and computation, vol. 4 (1994), pp. 499–528. - Patrick Dehornoy. From large cardinals to braids via distributive algebra. Journal of knot theory and its ramifications, vol. 4 (1995), pp. 33–79. - J. R. Steel. The well-foundedness of the Mitchell order. The journal of symbolic logic, vol. 58 (1993), pp. 931–940. - Randall Dougherty. Critical points in an algebra of elementary embeddings. Annals of pure and applied logic, vol. 65 (1993), pp. 211–241. - Randall Dougherty. Critical points in an algebra of elementary embeddings, II. Logic: from foundations to applications, European logic colloquium, edited by Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, pp. 103–136. - Randall Dougherty and Thomas Jech. Finite left-distributive algebras and embedding algebras. Advances in mathematics, vol. 130 (1997), pp. 201–241.

2002 ◽  
Vol 8 (4) ◽  
pp. 555-560
Author(s):  
Aleš Drápal
Keyword(s):  
New York ◽  
Critical Points ◽  
American Mathematical Society ◽  
Braid Groups ◽  
Alternative Proof ◽  
Elementary Embedding ◽  
Oxford University ◽  
Elementary Embeddings ◽  
Free Left ◽  
Distributive Law

Families of quasigroup operations satisfying the generalized distributive law

2020 ◽  
Vol 30 (3) ◽  
pp. 187-202
Author(s):  
Sergey V. Polin
Keyword(s):  
System Of Equations ◽  
Systems Of Equations ◽  
The Family ◽  
Distributive Law ◽  
Elimination Of Variables

AbstractThe previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

scholarly journals Certain characterizations of varieties of bands

1988 ◽  
Vol 31 (2) ◽  
pp. 301-319 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Rectangular Band ◽  
Simple System ◽  
Left Zero ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Lattice Of Varieties ◽  
Green's Relations ◽  
The World ◽  
New System

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Salvage of Total Esophageal Reconstruction with Microvascular Free Left Colon Flap

2002 ◽  
Vol 18 (4) ◽  
pp. 261-262 ◽  
Author(s):  
Chris D. Tzarnas ◽  
William S. Wittenborn ◽  
Joseph D. Whitlark ◽  
William H. Ayers
Keyword(s):  
Left Colon ◽  
Esophageal Reconstruction ◽  
Free Left
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