Page 273 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 273
VII: Further Topics in Skew Boolean Algebras
Theorem 7.1.11. PD0 is term equivalent to the variety of all right-handed skew Boolean
∩-algebras. Given (A; d, 0) in PD0, right-handed skew Boolean ∩-operations are given by
x ∨ y = d(x, 0, y), x ∧ y = d(x, d(x, 0, y), y) and x ⁄ y = d(x, y, 0).
Conversely, given a right-handed skew Boolean ∩-algebra (S; ∨, ∧, /, 0), an algebra (S; d, 0) in
PD0 is given by
d(x, y, z) = x / y ∨ (x ∧ z) ∨ (y ∨ z)/y.
Finally both processes, {d, 0} → {∨, ∧, /, 0} and {∨, ∧, /, 0} → {d, 0}, are reciprocal.
Proof. If (A; d, 0) is a pointed ternary discriminator algebra, then
d(x, 0, y) = ⎧x if x ≠ 0 ; d(x, d(x, 0, y), y) = ⎧y if x ≠ 0 ; d(x, y, 0) = ⎧0 if x = y .
⎩⎨y if x = 0 ⎩⎨0 if x = 0 ⎩⎨ x if x ≠ y
But these identities describe respectively ∨, ∧ and / on the primitive right-handed Boolean ∩-
algebra with upper D-class A \ {0} and zero-class {0}. Hence all algebras in PD0 induce right-
handed skew Boolean ∩-algebras. Conversely, given (S; ∨, ∧, /, 0) and d(x, y, z) as stated, then
d(x, x, z) = (x ∧ z) ∨ (x ∨ z)/x = z and d(0, y, z) = (y ∨ z)/y = ⎧0 if y ≠ 0 .
⎨⎩z if y = 0
When x is neither y nor 0, then d(x, y, z) = x ∨ (x ∧ z) ∨ (y ∨ z)/y = x. Thus in all cases d(x, y, z)
is indeed the ternary discriminator on S and (S; d, 0) is a pointed discriminator algebra. Thus all
algebras (S; d, 0) induced from right-handed skew Boolean ∩-algebras lie in PD0. That the
operations are reciprocal is easily checked at the “entry level” of pointed discriminator algebras
and primitive right-handed Boolean ∩-algebras. £
Corollary 7.1.12. Any skew Boolean ∩-algebra A has a right-handed skew Boolean ∩-
algebra reduct AR with the property that its congruences and its ∩, / and \ operations coincide
with those of A. The new skew lattice operations ∨R and ∧R are defined in terms of the old by
x ∨R y = x∨y∨x and x ∧R y = y∧x∧y.
Proof. Indeed one has
x ∨R y = d(x, 0, y) = (x / 0) ∨ [y \ ( (x / 0) ∨ (0 / x) )] = x ∨ (y \ x) = x ∨ y ∨ x
with the last identity holding first on all primitive ∩-algebras and hence on all ∩-algebras. Also,
x ∧R y = d(x, d(x, 0, y), y) = d(x, x ∨ y ∨ x, y) = y \ [(x ∨ y ∨ x) / x] = y∧x∧y
with the last identity holding again first on all primitive ∩-algebras and thus on all ∩-algebras.
That the standard differences agree is clear. Intersections also agree since the natural partial
ordering is unchanged: x∧y = y∧x = x iff x∧y∧x = y∧x∧y = x. Thus BCK differences are also
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Theorem 7.1.11. PD0 is term equivalent to the variety of all right-handed skew Boolean
∩-algebras. Given (A; d, 0) in PD0, right-handed skew Boolean ∩-operations are given by
x ∨ y = d(x, 0, y), x ∧ y = d(x, d(x, 0, y), y) and x ⁄ y = d(x, y, 0).
Conversely, given a right-handed skew Boolean ∩-algebra (S; ∨, ∧, /, 0), an algebra (S; d, 0) in
PD0 is given by
d(x, y, z) = x / y ∨ (x ∧ z) ∨ (y ∨ z)/y.
Finally both processes, {d, 0} → {∨, ∧, /, 0} and {∨, ∧, /, 0} → {d, 0}, are reciprocal.
Proof. If (A; d, 0) is a pointed ternary discriminator algebra, then
d(x, 0, y) = ⎧x if x ≠ 0 ; d(x, d(x, 0, y), y) = ⎧y if x ≠ 0 ; d(x, y, 0) = ⎧0 if x = y .
⎩⎨y if x = 0 ⎩⎨0 if x = 0 ⎩⎨ x if x ≠ y
But these identities describe respectively ∨, ∧ and / on the primitive right-handed Boolean ∩-
algebra with upper D-class A \ {0} and zero-class {0}. Hence all algebras in PD0 induce right-
handed skew Boolean ∩-algebras. Conversely, given (S; ∨, ∧, /, 0) and d(x, y, z) as stated, then
d(x, x, z) = (x ∧ z) ∨ (x ∨ z)/x = z and d(0, y, z) = (y ∨ z)/y = ⎧0 if y ≠ 0 .
⎨⎩z if y = 0
When x is neither y nor 0, then d(x, y, z) = x ∨ (x ∧ z) ∨ (y ∨ z)/y = x. Thus in all cases d(x, y, z)
is indeed the ternary discriminator on S and (S; d, 0) is a pointed discriminator algebra. Thus all
algebras (S; d, 0) induced from right-handed skew Boolean ∩-algebras lie in PD0. That the
operations are reciprocal is easily checked at the “entry level” of pointed discriminator algebras
and primitive right-handed Boolean ∩-algebras. £
Corollary 7.1.12. Any skew Boolean ∩-algebra A has a right-handed skew Boolean ∩-
algebra reduct AR with the property that its congruences and its ∩, / and \ operations coincide
with those of A. The new skew lattice operations ∨R and ∧R are defined in terms of the old by
x ∨R y = x∨y∨x and x ∧R y = y∧x∧y.
Proof. Indeed one has
x ∨R y = d(x, 0, y) = (x / 0) ∨ [y \ ( (x / 0) ∨ (0 / x) )] = x ∨ (y \ x) = x ∨ y ∨ x
with the last identity holding first on all primitive ∩-algebras and hence on all ∩-algebras. Also,
x ∧R y = d(x, d(x, 0, y), y) = d(x, x ∨ y ∨ x, y) = y \ [(x ∨ y ∨ x) / x] = y∧x∧y
with the last identity holding again first on all primitive ∩-algebras and thus on all ∩-algebras.
That the standard differences agree is clear. Intersections also agree since the natural partial
ordering is unchanged: x∧y = y∧x = x iff x∧y∧x = y∧x∧y = x. Thus BCK differences are also
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