Though this struck me as vaguely familiar when I found it, and very clearly mid-century, I didn't know about Richard Saar and Saar Ceramics--but am very happy to now, including to know that Saar was the husband of artist Betye Saar and father of Alison and Lezley! Launched in Lawndale, California and soon relocated to a small Quonset hut in El Segundo, Saar Ceramics studio operated from 1949 to 1962 producing tableware, tiles, and figural sculptures with forms and glaze designs inspired by indigenous cultures, nature, and mid-century modern abstraction. Richard Saar (1924–2004) designed  the forms and hand-painted the pieces, and his brother, William Saar, managed the business and assisted in the production. The American Museum of Ceramic Art organized a significant Saar Ceramics exhibition in 2018 curated by Jo Lauria--lots more there.  

Included in photos are several product sheets, including one featuring this specific form, described as a cigarette box with ashtray; quite a smart design, embodying the best of the work I think both in term of form and decoration--and equally useful for serving olives and then hiding the pits! It doesn't look to me as if this one ever saw an ash; it is is in pretty much mint condition and excellent treated as a box with a striking and useful tray top.  

Base: 4 x 6 1/2 x 1 5/16. Tray 8 x 5. Excellent condition, 1950s, signed on the underside of both tray and base.