Every quasitrivial n-ary semigroup is reducible to a semigroup

in Algebra Universalis (2019), 80(4),

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in ... [more ▼]

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲]

Galois theory for sets of operations closed under permutation, cylindrification and composition

in Algebra Universalis (2012), 67(3), 273-297

A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and ... [more ▼]

A set of operations on A is shown to be the set of linear term operations of some algebra on A if and only if it is closed under permutation of variables, addition of inessential variables, and composition, and if it contains all projections. A Galois framework is introduced to describe the sets of operations that are closed under the operations mentioned above, not necessarily containing all projections. The dual objects of this Galois connection are systems of pointed multisets, and the Galois closed sets of dual objects are described accordingly. Moreover, the closure systems associated with this Galois connection are shown to be uncountable (even if the closed sets of operations are assumed to contain all projections). [less ▲]

Uniqueness of minimal coverings of maximal partial clones

in Algebra Universalis (2011), 65(4), 393-420

A partial function f on a k-element set Ek is a partial Sheffer function if every partial function on Ek is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone ... [more ▼]

A partial function f on a k-element set Ek is a partial Sheffer function if every partial function on Ek is definable in terms of f. Since this holds if and only if f belongs to no maximal partial clone on Ek, a characterization of partial Sheffer functions reduces to finding families of minimal coverings of maximal partial clones on Ek. We show that for each k ≥ 2, there exists a unique minimal covering. [less ▲]