Semigroud

A semigroud or semiheap is an Algebraic Theory with a single Ternary Operation [ − , − , − ] : A → A → A → A : → 𝐴 𝐴 → 𝐴 → 𝐴 [-,-,-]:A\to A\to A\to A that is Para-associative; EG:

Examples

The canonical example of a semigroud is the type of Nonempty Lists, with [ x ⁢ s , y ⁢ s , z ⁢ s ] = x ⁢ s + reverse ⁢ ( y ⁢ s ) + z ⁢ s 𝑥 𝑠 𝑦 𝑠 𝑧 𝑠 𝑥 𝑠 reverse 𝑦 𝑠 𝑧 𝑠 [xs,ys,zs]=xs+\mathrm{reverse}(ys)+zs .

The type of Matrices is also a semigroud [ A , B , C ] = A ⁢ B T ⁢ C 𝐴 𝐵 𝐶 𝐴 superscript 𝐵 𝑇 𝐶 [A,B,C]=AB^{T}C .

Every hom set of an Allegory is a semigroud; this is a generalization of the fact that Relations form a semigroud with Composition of Relations and Converse of Relations.

Semigrouds and Involutive Semigroups

In general every Involutive Semigroup gives rise to a semigroud, with [ x , y , z ] = x ⁢ y † ⁢ z 𝑥 𝑦 𝑧 𝑥 superscript 𝑦 † 𝑧 [x,y,z]=xy^{\dagger}z . The previous two examples are instances of this phenomena.

In the reverse direction, every semigroud with a Biunitary Element e : A : 𝑒 𝐴 e:A forms an Involutive Semigroup, with a ⁢ b = [ a , e , b ] 𝑎 𝑏 𝑎 𝑒 𝑏 ab=[a,e,b] , and a † = [ e , a , e ] superscript 𝑎 † 𝑒 𝑎 𝑒 a^{\dagger}=[e,a,e] .

Moreover, every semigroud embeds into an Involutive Semigroup; see Wagner’s Theory of Generalised Heaps, 8.2.11.