Cancellation in Skew Lattices

被引:17
作者
Cvetko-Vah, Karin [2 ]
Kinyon, Michael [1 ]
Leech, Jonathan [3 ]
Spinks, Matthew [4 ]
机构
[1] Univ Denver, Dept Math, Denver, CO 80208 USA
[2] Univ Ljubljana, Dept Math, Ljubljana 1000, Slovenia
[3] Westmont Coll, Dept Math, Santa Barbara, CA 93108 USA
[4] Univ Bern, Math Inst, CH-3012 Bern, Switzerland
来源
ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS | 2011年 / 28卷 / 01期
关键词
Skew lattice; Cancellation; Distributivity; Variety; BOOLEAN-ALGEBRAS; RINGS;
D O I
10.1007/s11083-010-9151-7
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Distributive lattices are well known to be precisely those lattices that possess cancellation: x boolean OR y = x boolean OR z and x boolean AND y = x boolean AND z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M-3 or N-5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations. and. no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M-3 or N-5 that ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x boolean OR z = y boolean OR z and x boolean AND z = y boolean AND z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.
引用
收藏
页码:9 / 32
页数:24
相关论文
共 19 条
[1]  
Bignall R. J., 2003, SCI MATH JPN, V58, P629
[2]   SKEW BOOLEAN-ALGEBRAS AND DISCRIMINATOR VARIETIES [J].
BIGNALL, RJ ;
LEECH, JE .
ALGEBRA UNIVERSALIS, 1995, 33 (03) :387-398
[3]   Propositional skew Boolean logic [J].
Bignall, RJ ;
Spinks, M .
1996 26TH INTERNATIONAL SYMPOSIUM ON MULTIPLE-VALUED LOGIC, PROCEEDINGS, 1996, :43-48
[4]  
BIGNALL RJ, INT J ALGEB IN PRESS
[5]  
BIRKHOFF G, 1967, AMS C PUBLICATIONS, V25
[6]   Associativity of the del-operation on bands in rings [J].
Cvetko-Vah, Karin ;
Leech, Jonathan .
SEMIGROUP FORUM, 2008, 76 (01) :32-50
[7]   Internal decompositions of skew lattices [J].
Cvetko-Vah, Karin .
COMMUNICATIONS IN ALGEBRA, 2007, 35 (01) :243-247
[8]   A new proof of Spinks' Theorem [J].
Cvetko-Vah, Karin .
SEMIGROUP FORUM, 2006, 73 (02) :267-272
[9]  
CVETKOVAH K, 2005, THESIS U LJUBLJANA
[10]   NORMAL SKEW LATTICES [J].
LEECH, J .
SEMIGROUP FORUM, 1992, 44 (01) :1-8