Posetic Tools in the Social Sciences: A Tutorial Exposition
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8. Supporting employees with intellectual disabilities. Partial order theory has
been recently used to support the definition of the requirements, for the development of an assistive control panel and display interface to enable employees with
intellectual disabilities to extend their range of tasks and to increase their level of
responsibility (Fuhrmann et al. 2018).
All in all, what these examples show is that the concepts and the tools from
partial order theory allow for “classical” problems (like ranking or evaluation. . . )
to be consistently addressed in an ordinal setting, paving the way to deeper, more
reliable and more effective representations of complex socio-economic traits. This
should be enough to acknowledge the key role of partial order theory in supporting
decision-making processes in an increasingly complex world (for a more general
discussion, see Fattore and Maggino 2014).
2 A Few Technical Notes
In this section, we fix the terminology and collect some essential concepts of
partial order theory, also briefly touching upon available software resources, for
practical applications. More details and the mathematical proofs can be found in
cited references.
2.1 Basic Definitions
A partially ordered set (or a poset) π = (X, π ) is a set X endowed with a partial
order relation π , i.e. with a reflexive, antisymmetic and transitive binary relation
(Davey and Priestley 2002; Schröder 2016). Two elements x i and x j of the poset
are called comparable, if either x i π x j or x j π x i , otherwise they are called
incomparable (written x i || π x j ). A poset where any two elements are comparable
is called a linear order or a complete order or a total order. A subset of a poset is
called a chain if any two of its elements are comparable: at the opposite, it is called
an antichain, if any two of its elements are incomparable. The upset of an element
x i ∈ π , written x i ↑, is the set of elements dominating it: x i ↑= {x ∈ π : x i π x};
analogously, the downset of x i ∈ π , written x i ↓, is the set of elements dominated
by x i , i.e. x i ↓= {x ∈ π : x π x i }. Given x i , x j ∈ π , x j is said to cover x i (written
x i ≺ π x j ) if x i π x j and there is no other element x h ∈ π (x h = x i , x j ) such that
x i π x h π x j . If the poset has a finite number n of elements, the cover relation
determines the partial order relation, since x i π x j holds if and only if there exists a
sequence of elements x 0 , x 1 , . . . , x k , such that x i = x 0 ≺ π x 1 ≺ π . . . ≺ π x k = x j .
In the following, we consider only finite posets.
221
8. Supporting employees with intellectual disabilities. Partial order theory has
been recently used to support the definition of the requirements, for the development of an assistive control panel and display interface to enable employees with
intellectual disabilities to extend their range of tasks and to increase their level of
responsibility (Fuhrmann et al. 2018).
All in all, what these examples show is that the concepts and the tools from
partial order theory allow for “classical” problems (like ranking or evaluation. . . )
to be consistently addressed in an ordinal setting, paving the way to deeper, more
reliable and more effective representations of complex socio-economic traits. This
should be enough to acknowledge the key role of partial order theory in supporting
decision-making processes in an increasingly complex world (for a more general
discussion, see Fattore and Maggino 2014).
2 A Few Technical Notes
In this section, we fix the terminology and collect some essential concepts of
partial order theory, also briefly touching upon available software resources, for
practical applications. More details and the mathematical proofs can be found in
cited references.
2.1 Basic Definitions
A partially ordered set (or a poset) π = (X, π ) is a set X endowed with a partial
order relation π , i.e. with a reflexive, antisymmetic and transitive binary relation
(Davey and Priestley 2002; Schröder 2016). Two elements x i and x j of the poset
are called comparable, if either x i π x j or x j π x i , otherwise they are called
incomparable (written x i || π x j ). A poset where any two elements are comparable
is called a linear order or a complete order or a total order. A subset of a poset is
called a chain if any two of its elements are comparable: at the opposite, it is called
an antichain, if any two of its elements are incomparable. The upset of an element
x i ∈ π , written x i ↑, is the set of elements dominating it: x i ↑= {x ∈ π : x i π x};
analogously, the downset of x i ∈ π , written x i ↓, is the set of elements dominated
by x i , i.e. x i ↓= {x ∈ π : x π x i }. Given x i , x j ∈ π , x j is said to cover x i (written
x i ≺ π x j ) if x i π x j and there is no other element x h ∈ π (x h = x i , x j ) such that
x i π x h π x j . If the poset has a finite number n of elements, the cover relation
determines the partial order relation, since x i π x j holds if and only if there exists a
sequence of elements x 0 , x 1 , . . . , x k , such that x i = x 0 ≺ π x 1 ≺ π . . . ≺ π x k = x j .
In the following, we consider only finite posets.