てある details

write + (and…) + has been done and left that way

kaite aru

has been done and left that way (on purpose)

meaning: have been written

冷蔵庫れいぞうこにケーキが入いれてあるから、食たべていいよ。

reizouko ni keeki ga irete aru kara, tabete ii yo.

There's cake in the fridge for you — go ahead and have it. casual

会議かいぎの資料しりょうはもう印刷しいんさつしてある。

kaigi no shiryou wa mou insatsushite aru.

The materials for the meeting have been printed already. neutral

Examples

has been done (and remains so) — someone did it deliberately, and the result is what matters now

冷蔵庫れいぞうこにケーキが入いれてあるから、食たべていいよ。

reizouko ni keeki ga irete aru kara, tabete ii yo.

There's cake in the fridge for you — go ahead and have it. casual

会議かいぎの資料しりょうはもう印刷いんさつしてある。

kaigi no shiryou wa mou insatsushitearu.

The materials for the meeting have been printed already. neutral

お席せきに本日ほんじつの予定表よていひょうが置おいてあります。

o seki ni honjitsu no yoteihyou ga oitearimasu.

Today's schedule has been placed at your seat. formal

てある takes transitive verbs only, and the old object surfaces with が (or は): ドアが開けてある. When the doer stays topical, を survives: もう予約をしてある.

ている
窓が開いている just reports the state; 窓が開けてある adds a deliberate agent who left it that way — てある demands a transitive verb, ている takes any verb at all.

ておく
Same preparation scene, opposite viewpoint — 冷やしておく names the preparatory act, agent on stage and any tense available; 冷やしてある names the state that act left behind, agent already offstage.