# shifthorizontal

Let*i*be an integer between -7 and 7. The

*basic horizontal shift transform*

**U**

^{i}shifts a square on the chessboard

*i*squares right. (Directions are reversed if the argument is negative). If the square goes off the board, we say that

**U**

^{i}is

*empty*or

*undefined*.

A horizontal basic shift transform acts on square designators. For example,

because ifU^{3}a1 ≡ d1

`a1`

is shifted 3 squares right it becomes `d1`

.
As a special rule, suppose a square designator contains an entire rank, like `[a-h2,c4]`

contains the entire second rank in addition to the `c4`

square. Then any horizontal shifts of squares on that rank are ignored. Thus,

U^{3}[a-h2,c4] ≡ [a-h2,f4]

A horizontal shift transform can be extended to a horizontal shift transform of an arbitrary filter: each square designator appearing in the filter is shifted by the specified amount.

Sometimes a horizontal shift transform will result in an empty square designator. For example,

U^{7}[d2,c4]

will be empty. Any empty square designator makes the whole filter containing that empty square designator empty as well.

`shifthorizontal`

is a transform filter that represents each of the 15 basic horizontal shift transforms including the identity.

The effect of `shifthorizontal `

is first to create the 15 distinct filters that result from applying each of the basic horizontal shift transforms to *F**F*. Empty filters and repetitions are deleted, and the resulting set is called the *orbit* of `shifthorizontal `

.
*F*

For example, the orbit of `shifthorizontal ♔a1`

is 8 distinct filters, `♔a1`

, `♔b1`

, and so on.

On the other hand, the orbit of `shifthorizontal {♔a1 ♚g1}`

only contains 2 filters, namely `{♔a1 ♚g1}`

,
`{♔b1 ♚h1}`

. Any other horizontal shift results in one of the kings going off the board, and which makes the corresponding filter empty.

If *F* is a numeric filter then so is `shifthorizontal `

in which case its value is the maximum of the values of the matching transformed versions of *F**F*.

shifthorizontal countF

is a numeric filter whose value is the number of distinct matching transformed versions of *F*. For example, if the current position is the initial position, then

`shifthorizontal count find ♕a1`

is the number of different squares on the first rank on which there is a white queen in the game.