Page 138 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 138
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
The override and the update operations
J. Berendsen, D. Jansen, J. Schmaltz and F. Vaandrager in their 2010 paper, The
axiomatization of overriding and update (Journal of Applied Logic, 8 (2010), 141-150.), gave an
approach to studying partial function algebras that is similar in many ways to ours, but with some
differences. To begin, given a pair of partial functions f and g in the partial function set P(A, B)
with subsets F and G of A being their respective support, the authors considered the following
operations:
Override: f ▹ g = f ∪ g⎪(G \ F).
Update: f[g] = g⎪(F∩G) ∪ f⎪(F \ G).
Minus: f – g = f ⎪(F \ G).
The override f ▹ g is clearly the join f ∨ g, while the minus f – g is just f \ g. Stated in our
notation, the authors defined the update by f[g] = (f∧g) ∨ (f \ g) where f∧g is g \ (g \ f), the latter
holding for all right-handed skew Boolean algebras. For right-handed, strongly distributive skew
lattices in general, f[g] is given also as either (f∧g) ∨ f or f ∧ (g∨f). Indeed, using any of these
three ways, on 3R one has:
x[y] 0 1 2
0 0 0 0.
1 112
2 212
Thus all three evaluations of x[y] in any skew Boolean algebra agree. (The equation
(x∧y)∨x = x∧(y∨x)
and its ∨-∧ dual are examined in Section 5.1.)
One can consider algebras with signature ([], ∨) as was suggested by Berendsen et al in
their paper. The problem of interest for those authors was that of determining identities in [ ] and
∨ that hold for all partial function algebras (and thus more generally, for all right-handed,
strongly distributive skew lattices). Clearly all identities in [ ] and ∨ that hold in all partial
function algebras follow from the defining identities for right-handed skew Boolean algebras;
more generally, they follow from the defining identities for right-handed, strongly distributive
skew lattices. In their paper, the authors were interested particularly in finding a set of identities
in just [ ] and ∨ that (1) held for all partial function algebras and (2) was powerful enough so that
all identities in ∨ and [ ] that hold in all partial function algebras were consequences of the given
identities. The search for such a set of identities is apparently still open. However, if 3* denotes
the algebra ({0, 1, 2}; [ ], ∨) defined by the Cayley tables
x∨y 0 1 2 x[y] 0 1 2
0 012 and 0 000
1 111 1 112
2 222 2 212
136
The override and the update operations
J. Berendsen, D. Jansen, J. Schmaltz and F. Vaandrager in their 2010 paper, The
axiomatization of overriding and update (Journal of Applied Logic, 8 (2010), 141-150.), gave an
approach to studying partial function algebras that is similar in many ways to ours, but with some
differences. To begin, given a pair of partial functions f and g in the partial function set P(A, B)
with subsets F and G of A being their respective support, the authors considered the following
operations:
Override: f ▹ g = f ∪ g⎪(G \ F).
Update: f[g] = g⎪(F∩G) ∪ f⎪(F \ G).
Minus: f – g = f ⎪(F \ G).
The override f ▹ g is clearly the join f ∨ g, while the minus f – g is just f \ g. Stated in our
notation, the authors defined the update by f[g] = (f∧g) ∨ (f \ g) where f∧g is g \ (g \ f), the latter
holding for all right-handed skew Boolean algebras. For right-handed, strongly distributive skew
lattices in general, f[g] is given also as either (f∧g) ∨ f or f ∧ (g∨f). Indeed, using any of these
three ways, on 3R one has:
x[y] 0 1 2
0 0 0 0.
1 112
2 212
Thus all three evaluations of x[y] in any skew Boolean algebra agree. (The equation
(x∧y)∨x = x∧(y∨x)
and its ∨-∧ dual are examined in Section 5.1.)
One can consider algebras with signature ([], ∨) as was suggested by Berendsen et al in
their paper. The problem of interest for those authors was that of determining identities in [ ] and
∨ that hold for all partial function algebras (and thus more generally, for all right-handed,
strongly distributive skew lattices). Clearly all identities in [ ] and ∨ that hold in all partial
function algebras follow from the defining identities for right-handed skew Boolean algebras;
more generally, they follow from the defining identities for right-handed, strongly distributive
skew lattices. In their paper, the authors were interested particularly in finding a set of identities
in just [ ] and ∨ that (1) held for all partial function algebras and (2) was powerful enough so that
all identities in ∨ and [ ] that hold in all partial function algebras were consequences of the given
identities. The search for such a set of identities is apparently still open. However, if 3* denotes
the algebra ({0, 1, 2}; [ ], ∨) defined by the Cayley tables
x∨y 0 1 2 x[y] 0 1 2
0 012 and 0 000
1 111 1 112
2 222 2 212
136
