Definition 1.7 (Paths)

A path is a finite sequence of feature names, and the set Paths = Feats * Paths superscript Feats \mbox{\sc Paths}\ =\mbox{\sc Feats}^{*} Paths = Feats start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT is the collection of paths. We use π , α 𝜋 𝛼 \pi,\alpha italic_π , italic_α (with or without subscripts) to refer to paths. ϵ italic-ϵ \epsilon italic_ϵ is the empty path. The definition of δ 𝛿 \delta italic_δ is extended to paths in the natural way:

δ ⁢ ( q , ϵ ) = q δ ⁢ ( q , f ⁢ π ) = δ ⁢ ( δ ⁢ ( q , f ) , π ) 𝛿 𝑞 italic-ϵ 𝑞 𝛿 𝑞 𝑓 𝜋 𝛿 𝛿 𝑞 𝑓 𝜋 \begin{array}[]{l}\delta(q,\epsilon)=q\\ \delta(q,f\pi)=\delta(\delta(q,f),\pi)\end{array} start_ARRAY start_ROW start_CELL italic_δ ( italic_q , italic_ϵ ) = italic_q end_CELL end_ROW start_ROW start_CELL italic_δ ( italic_q , italic_f italic_π ) = italic_δ ( italic_δ ( italic_q , italic_f ) , italic_π ) end_CELL end_ROW end_ARRAY

The paths of a feature structure A 𝐴 A italic_A are Π ( A ) = { π ∣ π ∈ Paths fragments Π fragments normal-( A normal-) fragments normal-{ π normal-∣ π Paths \Pi(A)=\{\pi\mid\pi\in\mbox{\sc Paths} roman_Π ( italic_A ) = { italic_π ∣ italic_π ∈ Paths and δ ( q ¯ A , π ) ↓ } fragments δ fragments normal-( subscript normal-¯ 𝑞 𝐴 normal-, π normal-) normal-↓ normal-} \delta(\bar{q}_{A},\pi)\!\!\downarrow\} italic_δ ( ¯ start_ARG italic_q end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_π ) ↓ } .

Definition 3.1

𝒜 𝒜 \cal{A} caligraphic_A is called left-symmetric iff for all x , y , z ∈ 𝒜 𝑥 𝑦 𝑧 𝒜 x,y,z~{}\in~{}\cal{A} italic_x , italic_y , italic_z ∈ caligraphic_A the identity:

[ x , y , z ] = [ y , x , z ] , 𝑥 𝑦 𝑧 𝑦 𝑥 𝑧 \displaystyle[x,y,z]=[y,x,z], [ italic_x , italic_y , italic_z ] = [ italic_y , italic_x , italic_z ] ,

i.e.

( x ⋅ y ) ⋅ z - x ⋅ ( y ⋅ z ) = ( y ⋅ x ) ⋅ z - y ⋅ ( x ⋅ z ) ⋅ ⋅ 𝑥 𝑦 𝑧 ⋅ 𝑥 ⋅ 𝑦 𝑧 ⋅ ⋅ 𝑦 𝑥 𝑧 ⋅ 𝑦 ⋅ 𝑥 𝑧 \displaystyle(x\cdot y)\cdot z-x\cdot(y\cdot z)=(y\cdot x)\cdot z-y\cdot(x% \cdot z) ( italic_x ⋅ italic_y ) ⋅ italic_z - italic_x ⋅ ( italic_y ⋅ italic_z ) = ( italic_y ⋅ italic_x ) ⋅ italic_z - italic_y ⋅ ( italic_x ⋅ italic_z )

holds.