# Restriction in Program Algebra

@inproceedings{Jackson2021RestrictionIP, title={Restriction in Program Algebra}, author={Marcel Jackson and T. Stokes}, year={2021} }

intersection, ∩: f ∩ g = {(x, y) ∈ X × Y | (x, y) ∈ f and (x, y) ∈ g} and difference, \: f\g = {(x, y) ∈ X × Y | (x, y) ∈ f and (x, y) 6∈ g}. Before we proceed further, we note that the surrounding literature contains a number of conflicting notations for the operations just introduced. In [2], the operation we have denoted ◦ is referred to in an equivalent form as “intersection”, with definition and notation f@g := g ◦ f . The notation for ⊔ in [2] is ⊲, however, the notation ⊲ is well… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Comparison semigroups and algebras of transformations

- Mathematics
- 2010

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise.… Expand

The axiomatization of override and update

- Computer Science, Mathematics
- J. Appl. Log.
- 2010

It is established that override and minus are functionally complete in the sense that any operation on general functions that corresponds to a valid coloring of a Venn diagram can be described using just these two operations. Expand

Monoids with tests and the algebra of possibly non-halting programs

- Computer Science, Mathematics
- J. Log. Algebraic Methods Program.
- 2015

By extending the notion of {\em if-then-else}, this work is able to give finite axiomatisations of the resulting algebras of (partial) functions, with {\em while-do} in the signature if the state space is assumed finite. Expand

Modal restriction Semigroups: towards an Algebra of Functions

- Mathematics, Computer Science
- Int. J. Algebra Comput.
- 2011

This work considers restriction semigroups for which the usual Boolean operations on domains are modeled, leading to algebraic models of partial maps incorporating all the domain-related and set-theoretic operations previously considered. Expand

Difference-restriction algebras of partial functions: axiomatisations and representations

- Mathematics, Computer Science
- ArXiv
- 2020

The correctness of a finite equational axiomatisation for the class of algebras representable by partial functions with the signature of relative complement and domain restriction is provided and proved. Expand

Nearlattices with an Overriding Operation

- Mathematics, Computer Science
- Order
- 2011

A representation theorem (in terms of algebras of partial functions) for Boolean nearlattices with associative overriding is presented, and several results concerning associativity of the operation and the structure of function nearlATTices with overriding are presented. Expand

Relation algebras and function semigroups

- Mathematics
- 1970

The concept of relation algebra unifies many familiar notions from algebra (especially those of systems having “natural” models as groups, Boolean algebras etc.). The fundamental theorem on relation… Expand

Skew lattices and binary operations on functions

- Computer Science, Mathematics
- J. Appl. Log.
- 2013

It is shown the latter to be term equivalent to the variety of right-handed skew Boolean algebras, and both override and update operations are studied within the broader context of skew lattices with an eye towards achieving greater insight into their joint algebraic behavior. Expand

Skew Boolean algebras and discriminator varieties

- Mathematics
- 1995

We investigate the class of skew Boolean algebras which are also meet semilattices under the natural skew lattice partial order. Such algebras, called hereskew Boolean ∩-algebras, are quite common.… Expand

Override and update

- Mathematics, Computer Science
- ArXiv
- 2019

This work uses an unexpected connection with combinatorial geometry to provide a complete finite system of equational axioms for the first order theory of the override and update constructions on partial functions, resolving the main unsolved problem in the area. Expand