This chapter describes two Tcl/Tk graphical interfaces that can be used to define and edit semigroups and automata.

`‣ XAutomaton` ( [A] ) | ( function ) |

The function ` Xautomaton `

without arguments opens a new window where an automaton may be specified. A finite automaton (which may then be edited) may be given as argument.

gap> XAutomaton();

It opens a window like the following:

` Var `

is the ` States `

is the number of states, ` Alphabet `

represents the alphabet and may be given through a positive integer (in this case the alphabet is understood to be ` a,b,c,... `

) or through a string whose symbols, in order, being the letters of the alphabet. The numbers corresponding to the initial and accepting states are placed in the respective boxes. The automaton may be specified to be deterministic, non deterministic or with epsilon transitions. After pressing the A non deterministic automaton may be given as follows:

By pressing the button ** Ok ** the **GAP** shell aquires the aspect shown in the following picture and the automaton ` ndAUT `

may be used to do computations. Some computations such as getting a rational expression representing the language of the automaton, the (complete) minimal automaton representing the same language or the transition semigroup of the automaton, may be done directly after pressing the ** Functions** button.

By pressing the button ** View ** an image representing the automaton is displayed in a new window.

An automaton with epsilon transitions may be given as follows shown in the following picture. The last letter of the alphabet is always considered to be the ϵ. In the images it is represented by @.

A new window with an image representing the automaton may be obtained by pressing the button ** View **.

In the next example it is given an argument to the function `XAutomaton`

.

gap> A := RandomAutomaton("det",2,2); < deterministic automaton on 2 letters with 2 states > gap> XAutomaton(A);

It opens a window like the following:

The most common ways to give a semigroup to are through generators and relations, a set of (partial) transformations as generating set and as syntactic semigroups of automata or rational languages.

`‣ XSemigroup` ( [S] ) | ( function ) |

The function ` XSemigroup `

without arguments opens a new window where a semigroup (or monoid) may be specified. A finite semigroup (which may then be edited) may be given as argument.

gap> XSemigroup();

It opens a window like the following:

where one may choose how to give the semigroup.

In the window opened by `XSemigroup`

, by pressing the button **Proceed** the window should enlarge and have the following aspect. (If the window does not enlarge automatically, use the mouse to do it.)

` GAP variable `

is the When giving the relations, the usual abbreviations "0" and "1" may be used.

Pressing the **Done** button would output the following to the shell where **GAP** is running:

The menu button **Functions** has the following commands:

The interface allows to add and remove

By pressing the menu button **Functions** and selecting "Draw Schutzenberger Graphs" would pop up the following window:

By pressing the menu button

By pressing the menu button

`XSemigroup(poi3);`

would pop up the following window, where everything should be clear:

`XSemigroup();`

would pop up the following window, where we would select "Syntatic semigroup", press the **Proceed** button and then choose either to give a "Rational expression" or an "Automaton" by pressing one of those buttons:

If "Rational expression" is chosen, a new window pops up where the expression can be specified:

After pressing the

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