Fuzzy modal logics are a family of logics that are still under research for their under- standing. Several papers have been published on this issue, treating different problems about the fuzzy modal logics (see for instance [CR10], [CR11], [BEGR11], [HT06], [HT13], or [CMRR13]). However, the study of the product moda l logics, which we understand as logics that arise from Kripke structures whos e relation and universes are evaluated over the product standard algebra, has remained undone. We present here some results to partially fill that gap for Kripke semantics with crisp accessibility relations, together with a characterization of a strongly complete infinitary product logic with truth-constants. We consider that the characterization and understanding of the product modal logics could open the possibility of studying the more general case of BL modal logics

The authors acknowledge support of the Spanish projects EdeTRI (TIN2012-39348-C02-01) and AT (CONSOLIDER CSD 2007- 0022). A. V. is supported by a CSIC grant JAE Predoc.

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