Page 98 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 98
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Suppose next that we are just given aijθakl for some i < k. Then
ai1 = (aij∨a11) θ akl∨a11 = ak1.
Applying f(x) =(x ∧a11)∨a11 to both sides of ai1θak1, as often as needed, eventually yields
a11θa12 and hence θ = ∇.
Finally, suppose that ahiθahk with i ≠ k. If ahi and ahk lie in different rows in the
(∧)-array, then ahi∧ahk and ahk∧ahi are θ–equivalent, but have differing first indices, thus
returning us to the previous case to obtain θ = ∇. Otherwise, look at amm∧ahi θ amm∧ahk in the
final row of the ∧-array. If both ∧-products have differing first indices we again return to the
previous case. Otherwise, a1i = a11∧( amm ∧ ahi) θ a11∧( amm ∧ ahk) = a1kʹ. with i < iʹ and k < kʹ.
We repeat this cycle of calculations until a pair of θ-related elements with distinct first indices is
eventually encountered, which must happen due to the design of the ∧-array. We are then
returned to the previous case. £
By contrast, for any flat antilattice A, Con(A) is the lattice Equ(A) of all equivalences on
the underlying set of A. Flat antilattices of all types together generate the variety of regular
antilattices for which R(∨), L(∨), R(∧) and L(∧) are congruences. Such an algebra A factors as the
direct product of flat antilattices of each type, A ≅ A(l, l) × A(l, r) × A(r, l) × A(r, r) with Con(A)
correspondingly factoring as
Con(A) ≅ Equ(A(l, l)) × Equ(A(l, r)) × Equ(A(r, l)) × Equ(A(r, r)).
Flatness and regularity are studied in a broader context in the next section.
3.3 Regular quasilattices
Recall that a noncommutative lattice is flat if one of the following pairs of identities is
satisfied:
(r, l): a∨b∨a = b∨a and a∧b∧a = a∧b.
(l, r): a∨b∨a = a∨b and a∧b∧a = b∧a.
(l, l): a∨b∨a = a∨b and a∧b∧a = a∧b.
(r, r): a∨b∨a = b∨a and a∧b∧a = b∧a.
Thus, being (r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ.
Modified remarks hold for the other three types of flatness.
A noncommutative lattice is regular if R(∨) L(∨), R(∧) and L(∧) are all congruences, in
which case D(∨) and D(∧) are also congruences. While flat quasilattices are regular, this is not the
case for flat paralattices. In general:
96
Suppose next that we are just given aijθakl for some i < k. Then
ai1 = (aij∨a11) θ akl∨a11 = ak1.
Applying f(x) =(x ∧a11)∨a11 to both sides of ai1θak1, as often as needed, eventually yields
a11θa12 and hence θ = ∇.
Finally, suppose that ahiθahk with i ≠ k. If ahi and ahk lie in different rows in the
(∧)-array, then ahi∧ahk and ahk∧ahi are θ–equivalent, but have differing first indices, thus
returning us to the previous case to obtain θ = ∇. Otherwise, look at amm∧ahi θ amm∧ahk in the
final row of the ∧-array. If both ∧-products have differing first indices we again return to the
previous case. Otherwise, a1i = a11∧( amm ∧ ahi) θ a11∧( amm ∧ ahk) = a1kʹ. with i < iʹ and k < kʹ.
We repeat this cycle of calculations until a pair of θ-related elements with distinct first indices is
eventually encountered, which must happen due to the design of the ∧-array. We are then
returned to the previous case. £
By contrast, for any flat antilattice A, Con(A) is the lattice Equ(A) of all equivalences on
the underlying set of A. Flat antilattices of all types together generate the variety of regular
antilattices for which R(∨), L(∨), R(∧) and L(∧) are congruences. Such an algebra A factors as the
direct product of flat antilattices of each type, A ≅ A(l, l) × A(l, r) × A(r, l) × A(r, r) with Con(A)
correspondingly factoring as
Con(A) ≅ Equ(A(l, l)) × Equ(A(l, r)) × Equ(A(r, l)) × Equ(A(r, r)).
Flatness and regularity are studied in a broader context in the next section.
3.3 Regular quasilattices
Recall that a noncommutative lattice is flat if one of the following pairs of identities is
satisfied:
(r, l): a∨b∨a = b∨a and a∧b∧a = a∧b.
(l, r): a∨b∨a = a∨b and a∧b∧a = b∧a.
(l, l): a∨b∨a = a∨b and a∧b∧a = a∧b.
(r, r): a∨b∨a = b∨a and a∧b∧a = b∧a.
Thus, being (r, l)-flat means that D(∨) = R(∨) and D(∧) = L(∧), or equivalently, L(∨) = R(∧) = Δ.
Modified remarks hold for the other three types of flatness.
A noncommutative lattice is regular if R(∨) L(∨), R(∧) and L(∧) are all congruences, in
which case D(∨) and D(∧) are also congruences. While flat quasilattices are regular, this is not the
case for flat paralattices. In general:
96
