Filter Products Widget has check-boxes to select Categories by default as the default Display mode is set to Vertical list. There are three display modes (Vertical list, Column list, Dropdown list) available to choose from the settings (Edit »Settings »
Choose Display mode
from the drop-down list).
Vertical list and the Column list has the check-boxes to select the categories while the Dropdown list has the drop-down list to choose a single category which is similar of Radio button in the sense of choosing only single category. So to obtain OR logic instead of AND, Dropdown list
can be set to the Display mode
setting from the widget designer.
It is not possible to change the Filter products widget's behaviour and to obtain a customized logic it is better to implement a custom widget. Please read the documentation on how to create a custom widget here at http://docs.sitefinity.com/for-developers-create-a-new-custom-widget
. There it would be possible to add filter or dynamically generate/query the list through Sitefinity API (Ref. http://docs.sitefinity.com/for-developers-query-departments
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