Page 88 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 88
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Historical remarks
The material in Sections 2.1 and 2.2 appeared in Jonathan Leech’s 1989 paper in Algebra
Universalis. The material in Sections 2.3 and 2.6 appeared in his 1992 paper in the Semigroup
Forum. The material in Sections 2.4 and 2.5 originated in his 1993 paper in the Transactions of
the American Mathematical Society. Highlights from these papers appeared later in his 1996
survey article in the Semigroup Forum. An important result from his 1993 Transactions paper,
not presented here is the fact that the free symmetric skew lattice on two generators is infinite. As
a consequence, the free skew lattice on two generators must be infinite. Thus (symmetric) skew
lattices are not locally finite. This contrasts with the case for lattices where the free lattice on 2
generators has four elements, but the free lattice on three generators is infinite. In her 2011 paper
in Algebra Universalis, Karin Cvetko-Vah showed that for symmetric skew lattices that are also
categorical, the free algebra on two generators is finite of order 16. (But then thanks to the case
for lattices, the free such algebra on three generators must be infinite.) Her result echoed the case
for small skew lattices (≤ 2 generators) in rings that had been studied in Leech’s 2005 paper
below.
References
Jonathan Leech,
Skew lattices in rings, Algebra Universalis, 26 (1989), 48-72.
Normal skew lattices, Semigroup Forum, 44 (1992), 1-8.
The geometric structure of skew lattices, Transactions of the American Math. Society,
335 (1993), 823-845.
Recent developments in the theory of skew lattices, Semigroup Forum, 52 (1996), 7-24.
Small Skew Lattices in Rings, Semigroup Forum, 70 (2005), 307-311.
Karin Cvetko-Vah,
On strongly symmetric skew lattices, Algebra Universalis, 66 (2011), 99-113.
Internal decompositions of skew lattices, Communications in Algebra 35 (2006),
243 -247.
86
Historical remarks
The material in Sections 2.1 and 2.2 appeared in Jonathan Leech’s 1989 paper in Algebra
Universalis. The material in Sections 2.3 and 2.6 appeared in his 1992 paper in the Semigroup
Forum. The material in Sections 2.4 and 2.5 originated in his 1993 paper in the Transactions of
the American Mathematical Society. Highlights from these papers appeared later in his 1996
survey article in the Semigroup Forum. An important result from his 1993 Transactions paper,
not presented here is the fact that the free symmetric skew lattice on two generators is infinite. As
a consequence, the free skew lattice on two generators must be infinite. Thus (symmetric) skew
lattices are not locally finite. This contrasts with the case for lattices where the free lattice on 2
generators has four elements, but the free lattice on three generators is infinite. In her 2011 paper
in Algebra Universalis, Karin Cvetko-Vah showed that for symmetric skew lattices that are also
categorical, the free algebra on two generators is finite of order 16. (But then thanks to the case
for lattices, the free such algebra on three generators must be infinite.) Her result echoed the case
for small skew lattices (≤ 2 generators) in rings that had been studied in Leech’s 2005 paper
below.
References
Jonathan Leech,
Skew lattices in rings, Algebra Universalis, 26 (1989), 48-72.
Normal skew lattices, Semigroup Forum, 44 (1992), 1-8.
The geometric structure of skew lattices, Transactions of the American Math. Society,
335 (1993), 823-845.
Recent developments in the theory of skew lattices, Semigroup Forum, 52 (1996), 7-24.
Small Skew Lattices in Rings, Semigroup Forum, 70 (2005), 307-311.
Karin Cvetko-Vah,
On strongly symmetric skew lattices, Algebra Universalis, 66 (2011), 99-113.
Internal decompositions of skew lattices, Communications in Algebra 35 (2006),
243 -247.
86
