Page 24 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 24
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Given elements x and y, the product xy is the unique element in the row (R-class) of x and the
column (L-class) of y. Thus ek = g while ke = i in this array.
Returning to the broader context of arbitrary bands, we have
Corollary 1.2.5. Given any band S, its L-classes are the maximal left zero sub-bands in
S and its R-classes are the maximal right zero sub-bands in S. £
Theorem 1.2.6. (Clifford-McLean) Given a band S, the equivalence D is a congruence
on S. Its congruence classes form maximal rectangular sub-bands of S and the quotient algebra
S/D is the maximal semilattice image of S.
Proof. If e L f in S, then in general, ev L fv for all v in S. Thus for all u, v in S,
(uev)(ufv)(uev) = uevufvuev = uev(fv)ufvuev = uevfvuev = uevuev = uev
and similarly (ufv)(uev)(ufv) = ufv. Thus eLf implies uev D efv for all u, v in S. In like fashion,
eRf implies uev D efv for all u, v in S. Suppose that e D f in S. Then e R ef L f and thus for all
u, v in S, uev D uefv D ufv. D is thus a congruence. Since ef D fe for all e, f in S, S/D is
commutative and hence a semilattice. Since each D-class is a maximal subset of S satisfying
xyx = x, it is a maximal rectangular subalgebra of S. £
In brief, every band is a semilattice of rectangular bands. Thus a band has the appearance
of a semilattice diagram with each node filled in by a rectangular band.
££
££
£
While multiplication is performed in rectangular fashionwithin D-classes, multiplication between
elements from distinct D-classes is another matter. We can, however, be more specific in this
regard for the two subvarieties that we consider next.
Unlike D, the relations L and R need not be congruences. A band for which L and R are
full congruences is called regular. Since D is a congruence, whenever D = L (so that R = Δ) or
D = R (so that L = Δ) the band must be regular. When D = L the band is called left regular and
when D = R it is called right regular. It is both precisely when it is a semilattice. A normal, left
(right) regular band is called left (right) normal band.
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Given elements x and y, the product xy is the unique element in the row (R-class) of x and the
column (L-class) of y. Thus ek = g while ke = i in this array.
Returning to the broader context of arbitrary bands, we have
Corollary 1.2.5. Given any band S, its L-classes are the maximal left zero sub-bands in
S and its R-classes are the maximal right zero sub-bands in S. £
Theorem 1.2.6. (Clifford-McLean) Given a band S, the equivalence D is a congruence
on S. Its congruence classes form maximal rectangular sub-bands of S and the quotient algebra
S/D is the maximal semilattice image of S.
Proof. If e L f in S, then in general, ev L fv for all v in S. Thus for all u, v in S,
(uev)(ufv)(uev) = uevufvuev = uev(fv)ufvuev = uevfvuev = uevuev = uev
and similarly (ufv)(uev)(ufv) = ufv. Thus eLf implies uev D efv for all u, v in S. In like fashion,
eRf implies uev D efv for all u, v in S. Suppose that e D f in S. Then e R ef L f and thus for all
u, v in S, uev D uefv D ufv. D is thus a congruence. Since ef D fe for all e, f in S, S/D is
commutative and hence a semilattice. Since each D-class is a maximal subset of S satisfying
xyx = x, it is a maximal rectangular subalgebra of S. £
In brief, every band is a semilattice of rectangular bands. Thus a band has the appearance
of a semilattice diagram with each node filled in by a rectangular band.
££
££
£
While multiplication is performed in rectangular fashionwithin D-classes, multiplication between
elements from distinct D-classes is another matter. We can, however, be more specific in this
regard for the two subvarieties that we consider next.
Unlike D, the relations L and R need not be congruences. A band for which L and R are
full congruences is called regular. Since D is a congruence, whenever D = L (so that R = Δ) or
D = R (so that L = Δ) the band must be regular. When D = L the band is called left regular and
when D = R it is called right regular. It is both precisely when it is a semilattice. A normal, left
(right) regular band is called left (right) normal band.
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