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It seems that you in your INNER JOIN, you are joining on
wp_term_relationships.object_id == wp_posts.post_title
which is erroneous, since object_id is the post-id in this case-scenario.
I want all IDs (post-ids)
SELECT ID FROM $wpdb->posts
Who have a record in the wp_term_relationships table
// Now we are comparing the correct columns INNER JOIN $wpdb->term_relationships ON $wpdb->term_relationships.object_id = $wpdb->posts.ID
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Adding the additional where clause
Where the term_taxonomy_id is equal to 2
WHERE $wpdb->term_relationships.term_taxonomy_id = 2